Basic Attacks were meant to be the simplest form of combat in D&D 4th Edition...
...but ended up being one of the most complex, complicated, and tangled aspects of the game.
I'll try to untangle the mess here in this answer, but the simple answer is that a melee basic attack and a ranged basic attack are attacks every CREATURE can use. In other words, every player character, every NPC, and every monster can all make basic attacks, adding a certain layer of common attack across the entire game.
WARNING: What follows is largely based on analysis, synthesis, and assumption.
D&D 4e launched with the expectation of only one Melee Basic Attack.
The original Player's Handbook defined a
Melee Basic Attack as an attack power everyone possesses, regardless of class, and is, for everyone, a strength-based attack. Likewise, the original Player's Handbook expected Strength to govern all melee basic attacks always: all classes were expected to use strength to make melee basic attacks, including classes that used other stats for other melee attack powers.
This led to, for example, Rangers being split into dexterity-based archers and strength-based dual-wielders, Clerics being split into wisdom-based spellcasters and strength-based militants, Paladins being split into charisma-based divine defenders and strength-based swordsmen, and strength-based ruffianism being promoted amongst otherwise dexterity-based rogues.
In other words, there was originally a design idea that, no matter what you did in terms of your class, when it came to just plain hitting someone with a weapon, you did it with your strength.
The other main reason for the Melee Basic Attack was that the designers of D&D knew they were constructing an additive system: more and more at-will powers would be added with potentially unexpected functions; by introducing the Melee Basic Attack the designers thought they would have a constant and reliable tool around which to base certain mechanics.
D&D 4e set an early precedent for Melee Basic Attacks being important.
PHB1 had four particularly important uses for the Melee Basic Attack:
The expectation was that players would primarily use the Melee Basic Attack after using the charge action, making it important to players that needed to cover large distances or otherwise wanted the +1 bonus to attack.
The expectation was that players would primarily use the Melee Basic Attack as an Opportunity Attack. The premise was that provoking an opportunity attack would result in some damage but otherwise avoid applying the potentially crippling effects that would accompany the complex profusion of at-will powers. However, this had the consequence of rendering opportunity attacks made by characters that did not invest in strength meaningless, as they were unlikely to hit and, even if they did, not deal much damage. The rogue might be particularly deadly in melee, flanking with its dagger, but without strength monsters could just walk away without fear of the OA reprisal.
The Warlord introduced the concept of "lazy" play, or a character that grants other characters basic attacks. Originally, this meant a simple strength vs AC attack and powerful through its versatility rather than through the power being invoked. For example, granting a Melee Basic Attack to a Fighter would let that Fighter
mark an additional target per round; granting the attack to a Rogue, Ranger, or Warlock that had missed on their turn would let them invoke their Striker damage feature (Sneak Attack, Quarry, or Curse respectively; note these were all originally once-per-round effects).
The Fighter's mark reprisal feature, Combat Challenge, granted a Melee Basic Attack, in comparison to the other PHB1 defender, the Paladin, who's Divine Challenge dealt its own damage. A Fighter's ability to defend therefore relies on having a good Melee Basic Attack.
D&D 4e set an early precedent for alternate RANGED Basic Attacks.
Although there was only one Melee Basic Attack (strength vs AC), there were four Ranged Basic Attacks. The two weapon-based Ranged Basic Attacks were based on dexterity for light thrown and ranged weapons, and strength for heavy thrown weapons. Furthermore, Warlocks (Eldritch Blast) and Wizards (Magic Missile) had two at-will powers that specifically stated that they could be used as Ranged Basic Attacks.
On the other hand, there were no primary uses for Ranged Basic Attacks in PHB1: the Warlord could only grant Melee Basic Attacks at-will, and you couldn't use Ranged Basic Attacks on charges or Opportunity Attacks or Combat Challenge. This left PHB1 with an odd dichotomy of design, but also set the precedent for what would follow...
PHB2 introduced the way of swapping out Strength on the Melee Basic Attack.
PHB2 introduced the Melee Training feats, which allowed a player to change the attribute they used for attacking and damage with their Melee Basic Attack. This could be capitalized upon by a whole host of classes who had at-will melee attack powers but otherwise did not use strength, such as the newly introduced Avenger and Bard. This also made sense for the PHB1 Rogue and, to a lesser extent, the Paladin.
This was largely introduced to provide viable opportunity attack and charge attacks to these classes, especially Rogues and Avengers. Why would a rogue use dexterity to stab with its dagger on 99% of its attacks, but not when making a basic attack? Same with an Avenger using wisdom? The Melee Training feat was a patchwork solution to the odd logic of the original rules and seen by some as a math fix (not unlike Expertise, also introduced in PHB2) to be given for free to patch things up as needed.
PHB2 introduced the Druid, which needed its own unique Melee Basic Attack.
The Melee Basic Attack, as originally written, is a weapon attack; it was assumed that all characters under all circumstances would be able to make weapon (or unarmed) attacks. That all changed with the introduction of the shape-shifting Druid class in Player's Handbook 2.
The Druid's Wild Shape power limits the Druid to only using powers with the
Beast Form keyword: this rules out Melee Basic Attack. Such transformed Druids would lose all opportunity attack or charge abilities, so the authors helpfully supplied Druids with two at-will powers that could be used as Melee Basic Attacks (Savage Rend and Grasping Claws). Note that the third PHB3 Beast Form at-will, Pounce, could be used as a Melee Basic Attack but only on a charge.
The rules for building a Druid requires the player to pick at least one at-will power with the Beast Form keyword. Thus, while it was still possible for a Druid to have only Pounce and not be able to make opportunity attacks while in beast form, the Druid would at least always be able to charge.
Of particular note is that both of these powers also added additional functionality to put them in line with "normal" at-will powers: Savage Rend also slides the target 1 square and Grasping Claws also slows the target. This was an unusual decision: why didn't the designers simply add a "Beast Form Melee Basic Attack" that mirrored the typical Melee Basic Attack and grant it to Druids? Then they could have avoided adding special rules to the other beast form attacks...
In any case, these powers were not terribly problematic because although they replaced the Melee Basic Attack, they also required the
Beast Form keyword, preventing their use by weapon-wielding characters (but just wait...).
The Power books introduced more new Melee Basic Attacks.
For example, Arcane Power introduced a melee alternative for the Warlock's Eldritch Blast, Eldritch Strike. Presumably because Eldritch Blast was a Ranged Basic Attack, Eldritch Strike was made a Melee Basic Attack.
However, in addition to being a charisma-based or constitution-based rather than strength-based, Eldritch Strike also had the feature of sliding the target by 1 square on a hit. This was now a weapon-based attack that added considerable functionality to opportunity attacks and charges that didn't exist before.
Most frighteningly, if a Fighter used this power when their Combat Challenge was invoked, they could use an Immediate Interrupt to slide the attacker away from their target, negating the attack entirely as well as dealing damage; this was not only well beyond the original scope of the feature, it was also totally awesome and immediately focused upon by character optimizers.
As more material was published, Melee Basic Attacks became increasingly exploitable.
I won't go into the whole analysis of every issue, but as more and more things were added, and because of the nature of D&D that all things can be creatively combined by a sufficiently creative player with a large enough toolbox of options, Melee Basic Attacks took on a whole life of their own.
As a random example, Dragon Magazine introduced three Were- themes: Werewolf, Werebear, and Wererat. All three had a beast form transformation not unlike the Druid's Wild Shape; in fact, all three used the exact same
Beast Form keywords. Most interestingly, at level 10 characters with the Were- themes can transform into a half-humanoid half-beast hybrid, and can now use all powers with and without the
Beast Form keyword. This meant that a character could "poach" the Druid's Savage Rend and use it by half-shifting.
There are numerous other odd results of these features mixing together to create incredibly powerful Melee Basic Attacks that could be used on charges, on opportunity attacks, for the Fighter's combat challenge, and so forth.
Essentials made Basic Attacks Basic again...
Partway through D&D 4th Edition's lifespan, Wizards decided the game was too complex to attract new players and released a revised set of character options under the Essentials line: Heroes of the blah-blah.
Essentials tried to simplify all the rules by making the Basic Attack an explicit and central part of the character. Characters would explicitly have their Basic Attack and powers would modify that attack rather than give them new attacks.
For example, the Essentials Sorcerer has Elemental Bolt, a Ranged Basic Attack, and their Encounter power is Elemental Escalation, which makes Elemental Bolt hit an extra target and do extra damage.
This meant for much simpler play: the Sorcerer player simply uses the same power all the time: whenever they shoot, whenever they're granted basic attacks, whenever they augment with their Encounter power, and so forth.
At the same time, Essentials also vastly simplified rules surrounding Basic Attacks. They were now modified primarily through Stances, which became very common (many Essentials classes have two or more stances that modify their Basic Attack). The notion of granted attacks was done away with, as was power poaching: Half-Elf lost Dilettante, all multiclass feats were removed, and there were no rules for hybrid characters.
So, on the one hand, defenders now made MBAs as Opportunity Actions (rather than Immediate actions) and had a marking aura around them; on the other hand, they could never change their core class MBA so the designers always knew exactly what mark enforcement was being dished out. This led to a different, but internally balanced, style of play.
...but Essentials didn't mix well with the rest of D&D 4e
The problem with Essentials is that, while one part of Wizards of the Coast was trying to change fundamental design concepts to create a simpler style of game, another part of Wizards of the Coast was waving the "it's not a new game, it'll all work together" flag. This led to a number of diverging ways of delivering D&D to players: the official Online Character Builder lets you build pure Essentials characters, but Dragon magazine also published articles for making hybrid and multiclass Essentials characters to mix them with non-essentials characters.
This opened a huge can of exploitable worms because Essentials and "Vanilla" were built on very different fundamental concepts. In Essentials, the core idea of what a Basic Attack was intended to be changed. Rather than a very simple and common and function-free source of damage to be triggered on a number of specific circumstances, it became the primary tool around which the entire class was built.
This leads to a number of well-known exploits, such as taking an Essentials Pyro Sorcerer and buffing the fire-based Ranged Basic Attack with items such as Eagle Eye Goggles, which adds +1 to ranged basic attacks. Obviously, when the Goggles were made, it never occurred that a Ranged Basic Attack would actually be the most powerful power (or the only power) a class would have. It was never meat to be an item that plainly boosted the accuracy of an entire class's repertoire of (one) power.
Or, for example, poaching a powerful MBA and an Essentials Defender Aura.
Current state of Basic Attacks.
It's hard to really pin down the current state of Basic Attacks beyond "it's a whole load of messiness most DMs don't want to have to deal with." The state of character optimization definitely needs to be something the party discusses and agrees upon together, and modifications to Basic Attacks should be part of that discussion.
One thing that should be kept in mind is establishing a level playing field with all the players at the table. If some members of the party build convoluted characters to exploit Melee Basic Attacks and the others don't, combat can become a complex mess of monsters merrily trotting past worthless unoptimized opportunity attacks to become completely crippled by a well-executed (and optimized) basic attack.
My personal suggestion is to snuff out as much basic attack optimization as possible unless the entire group is prepared to handle them, and instead blanket convert the standard Melee Basic Attack rules to apply the same to all characters: if the character can make a melee attack at-will, then they use those values for to-hit and damage but apply no other effects on hit. For example, if there's a Rogue with Deft Strike, let them use the same to-hit and damage values as Deft Strike on a Melee Basic Attack (ie: dexterity) but without the 2 squares of movement.
This allows for simple and standardized opportunity attacks and charge attacks across all players, and lets everyone feel like they can stop monsters from just waltzing by willy-nilly.
As far as what you do in your campaign, that's up to you, but hopefully I've provided enough background here to make the right sort of choices that best fits your group :)