It takes a lot of time and effort to do it properly
It is difficult to mess up a single-class character1 but multiclassing efficiently is hard.
You have to keep in mind what features can be fueled by another class's resources (you can use Sorcerer spell slots for Divine Smite) and what not (Eldritch Strike only works with Warlock slots), what weaknesses can be overcome by multiclassing (Wizards can cast in Plate if proficient) and what not (Barbarians don't benefit from rage in Plate, even with proficiency).
You don't have to learn everything by heart, you can read the excellent guides out there, but that, again, takes a lot of time.
With careful planning, the downsides are negligible compared to the benefits
Some of the costs you might pay anyway (Ability Scores), and some are less important than they seem at first.
I have been playing some version of DnD since 1993, my highest level character was 15. Either a TPK forced a new start, or too many participants lost interest, and the game just died away.
Even if you reach level 20, it is just 5% of your career, how could it compete with Full Plate and a Shield for 19 levels?
Compare every level
Someone mentioned in one of the comments that while the +1d6 from Sneak Attack helped a lot his Fighter/Rogue in low levels, it was painful to watch how much stronger single-class characters were at level 5.
While this is to be expected, most likely he was stronger on levels 2-4, 6-10 and 12-19. So I would argue he was stronger overall. On those 3 levels hopefully the other players can help out, but this is the realm of party optimization.
You must have 13 in both Dex and Wis to multiclass into Monk, but I think it is very hard to create a decent Druid without these anyway, so this is not an extra cost.
Of course a Paladin/Monk is harder to achieve, you need a 13 in Str, Dex, Wis and Cha, but it is far from a great idea anyway.
It is true you do not get higher level spells for a long time even if you multiclass into another prime caster. However, spell power is not linear with spell level.
In the 11th level party I play in the most used spells are:
- Faerie Fire (because the target does not get a new save at the end of its turn)
- Blindness (because no Concentration)
- Hold Person (because criticals for the whole party)
- Bless (because no save, so always works)
Focus vs Versatility
Saying multiclassing provides versatility at the cost of focus is overly simplistic, and plain false in most cases.
A bad workman always blames his tools
If your Paladin spends all his ASIs on Intelligence, and you end up with a weak character, the problem is not with the concept of ASIs.
If your Wizard takes all the feats in alphabetical order, feats in general are not at fault.
Similarly, you should not blame multiclassing if a Fighter 10/Wizard 10 is weaker than an Eldritch Knight 20. This is just bad design, forgetting that level 11 is a big one for the Fighters. A F12/W8 would me much stronger.
It is entirely possible to become more focused by multiclassing:
- A Shield Master Fighter with one level of Rogue gets more out of the prone condition with Sneak Attack, and adding Expertise to Athletics makes him better at proning
- One level of Monk on a Moon Druid makes him better at Wild Shape, providing much needed AC to the beast forms (Unarmored Defense)
- Two levels of Paladin on a Moon Druid helps to spend the spell slots that would not see much use otherwise (Divine Smite)
- One or two levels of Fighter is a great addition for a Bladesinger, as Fighting Style, Second Wind and Action Surge can make you a better Warrior Mage
When to Multiclass
This depends heavily on the primary and secondary class, subclass and even on your Ability Scores.
When to stay:
- As others pointed out, level 5 a big jump for most classes, (not so much for Bards, Rogues and Moon Druids)
- Ability Score Improvements (ASI) are very important, so try to reach level 4 or 8 in all your classes
When to leave:
- Extra Attack does not stack with itself, so take it only from one class. A Barbarian 4/Fighter 6 is significantly stronger than a 5/5 split
- Paladins get their arguably strongest feature on level 11, I'd stay one more for ASI, but this is a very good place to get out
- Clerics get one of their best spells at level 7, stay one more for the ASI and the Domain feature, and switch to some other spellcaster for more spell slots or to a class with Extra Attack
- Warlocks learn a new spell every level up to 9, after that it is only once per two levels. Mystic Arcanum is also significantly weaker than getting new spell slots. Bladelocks might want to stay until level 12 for Lifedrinker, others should leave even sooner.
When to Dip
Many classes are front-loaded, meaning that some very good features are available at first or second level.
If you take so few from another class it is almost inconsequential when you do it, the most important thing to keep in mind you only get the saving throw proficiencies from your first class.
What to pick
- One level of Monk gives great Unarmored Defense for Druids and Clerics
- Two levels of Warlock gives a much needed powerful cantrip for Sorcerers and Bards (Eldritch Blast + Agonizing Blast)
- One level of Cleric gives casters Medium Armor Proficiency (possibly Heavy) and Domain features, with full spell progression
- One level of Fighter gives casters Armor Proficiency, Constitution Save and Second Wind
- Two levels of Fighter provides the only way to cast two spells in one round (Action Surge)
- One level of Rogue gives you Expertise if you want to be really good in two Skills, and +1d6 damage with some very easy conditions
- Two levels of Rogue gives you great mobility for bonus actions
Biggest drawback: Time
Bookkeeping and careful planning takes a lot of time:
- I have a Bard2/Sorcerer4/Warlock5 character, keeping track of spells known is a real pain, I had to open a spreadsheet for it
- A friend really wanted to add a few levels of Rogue to his Ranger, but realised too late that it would need a 13 in Wisdom too
TL;DR It is possible to create multiclass characters that are ahead of single-class characters in power, versatility or both. It just takes several hours of careful research and planning.
Read the guides on giantitp.com or enworld.com.
1 Unless you create a Beast Master Ranger, or a Way of the Four Elements Monk.