Stats for purchased dogs
The dog available for purchase from a kennel, as listed in the equipment chapter, has the statistics of a war dog in the Monster Manual or Monstrous Manual under Dog, and is a little bit beefier than a wild dog. The difference between the three is in what they are trained to be able to do, not in their statistics.
Dogs don't gain levels
Dogs are loyal companions, vigilant guards, and effective tools — but as far as the game's mechanics bother themselves with dogs, they're merely tools: extensions of their master's goals and plans. Just like a loyal warhorse doesn't advance no matter how many battles they've been in, a guard dog always stays the guard dog it was. Unlike the intelligent species, a dog is limited by its nature. An elven ranger may become a legendary archer, but a dog will never become a legendary... biter? They just don't advance.
This is possibly unfamiliar if you're coming from games where every last mouse can level up, but it's one of the major advantages of playing AD&D 2nd edition: creatures that aren't PC-grade are extraordinarily straightforward to use, whether you're playing at the low power of the first few levels or the high-flying epic end-game with 20th-level PCs. Monsters, animals, and so on don't "track" the PCs' power and level along with them.
How does a dog survive, then?!
If its master is careless, a dog doesn't. Dogs and other owned animals surviving, in 2e, is entirely not the DM's problem. If there are griffins flocking around and the PCs leave their horses staked outside the dungeon in the open, that just means the griffins eat well today. If a ranger sics loyal Barksmith McBitey on a manticore, the ranger has merely offered the manticore an appetizer.
There are a few ways to make dogs more survivable though:
They can be fitted with barding.
It's expensive, but once again that's the player's problem — if they don't think the high price of dog armour is worth the minor boost in AC, that is their own cost/risk analysis to make.
The cost of barding in the equipment lists is for horses. Feel free to make up your own prices for dogs, since they're probably much cheaper to bard. I'd personally go with ⅓ or ¼ the price of horse barding; those are just pulling numbers from a hat because you can't really go wrong here. Just like you make up situations for your players to deal with in an adventure, just pick something well-defined and let your player(s) decide what is and isn't worthwhile doing.
Buy two or ten more.
There's safety in numbers, and a pack of dogs is a vicious opponent even for a much more nasty enemy. Remember the overbearing rules! They're simple rules (PHB p. 98), and provide a way for a pack (or any mob of creatures) to knock down a single target and get them vulnerable on the ground. A particularly nasty target might still kill one or two dogs from a pack, but they would be wiser to run than to let themselves be surrounded and brought down.
(A reminder to the DM here: play both dogs and enemies like they value their lives. Few things fight to the death, or even the pain — even monsters. Running suicidal monsters is a sure way to limit a player's tactical options in combat, and make combats harder and bloodier than they need to be. Also use the Reaction roll rules and the Morale rules religiously!)
Dogs are replaceable.
Technically this doesn't make a single dog more survivable, but dogs are a fungible resource and effectively "immortal" because of that. It's personally sad when a dog dies, but a working dog can and will be replaced. Paying for Barksmith McBitey IV is like a resurrection spell for Barskmith III, but massively cheaper.
Dogs can be left behind when they're going to be outmatched.
Just like you don't take a sword to a gunfight, you don't take a dog into a dragon's lair. The life expectancy of a dog that is brought to bear wisely will be much higher than one thoughtlessly thrown into melee with any and every thing the PCs run into. Again, this is on the player's shoulders: let them figure out their own strategy, based on their experience with when a dog is and isn't useful. Give them opportunities to learn this too, by not sparing the life of a dog that has been put in lethal danger. Players can only learn when they have data, and in the life-or-death category of lessons, what results in life and what in death is necessary data.
Can dogs never get better, then?
They can actually! A dog may not have the intellectual and spiritual capacity to become legendarily competent via adventuring the way humanoids with classes can, but via training with the non-weapon proficiency Animal Training (Dog), a dog can be taught new tasks, behaviours, and tricks. It's a lot of investment in time (or money, if you're not training them personally) for only a very little bit of lateral improvement rather than upward improvement, but again, it's up to your player(s) to make these kinds of cost/reward assessments, not you.
You can do this two ways, and it's mostly up to your own sense of aethetics.
Aesthetic: Working animals are extensions of their master.
Don't give a thought to the XP. The player is paying out of pocket for a living "piece of equipment" that's making their job easier. That's just clever play, and the player should be awarded commensurately with XP for their PC. The PC gets full XP; regardless of how many horses, dogs, or hunting falcons they bring to bear in combat, they don't reduce the XP awarded.
Aesthetic: The dog makes challenges easier, so the PC shouldn't get full XP.
If you feel this way, then count the dog as a henchman in terms of "sucking away" XP from the player character(s). That means the dog takes up a ½ share of the XP when you divide it up. So if there's one ranger PC and one dog, the XP gets divided three ways and the ranger gets two of those shares. The dog "gets" one share. If there is one ranger and four dogs, the XP gets divided 1 + 4 × ½ = 3 three ways, and the ranger gets one of those shares. The rest go to the dogs.
A dog still doesn't advance with the XP it receives this way; this is just a mechanism to reduce the XP reward to the PCs according to how much help they got.
The costs of dog ownership
Owning one or more dogs is expensive, especially if you want them healthy for difficult work like guarding and fighting. They need a lot of meat, ready access to clean water, and regular care to keep them free of parasites internal and external. Injuries need tending, intra-pack dominance fighting needs to be curtailed (or paid for with more tending to wounds), constant reinforcement of training is required.
All this is time and money. Abstracting this away into a cost-of-living increase is a very sensible method of handling it. Though, if the player wants to engage directly with the expenses during play, that's something you should allow for. Especially with a ranger, the impact of caring for the dogs while in the wilderness will probably be significantly reduced simply by regularly bringing down a deer or two for them.
Aside: dogs are fun to DM
An NPC companion can be a pain to DM, because they represent not only a full personality to constantly develop and portray, but "should" also be smart and helpful and players can be very tempted to rely on their advice instead of thinking through the challenges of the game.
Non-verbal animals accompanying the PCs don't have this problem, because they can't talk! However, they still present the opportunity to convey information to the players, but it's up to the players to figure out what the non-verbal communication means and what to do about it. No player is going to have their PC ask McBitey for advice, which frees you up to be as communicative as possible with barks, movement, and described body language.
I personally find portraying non-verbal behaviours of animals quite a lot of fun. They aren't human, and don't think like humans. They have limited understanding of their surroundings and ability to communicate it, but they can provide a strong "early warning" system for a PC who is using them cleverly. You can push information as hard as you like with an animal because you can't really give away too much through the limited channels available to an animal.
And, it's not really important that you succeed in communicating — regardless of whether the player "gets" what the animal is trying to "say", you're engaged in an interesting spot of roleplaying, and the players have an opportunity (and notification) to take action: either gathering more information, making educated guesses as to what might be the matter, or both. Regardless, an animal wigging out about something prompts the players to react, and moves the game along, without leading them by the nose.