Monster spawning is likely to be some form of a
weighted random choice in most games. Well, I'm not sure how many games use it explicitly, but it's likely to be there in some form, and the method is general enough that at least explaining it should be a good way to get you started. The obvious implementation is putting all of the options in a list and iterating over that list twice (
one example I found by googling), which should be easy enough to write and more than fast enough unless you have hundreds of thousands of monster types in your game.
Using weighted random choice reduces your question to
what should be the probability of each monster type spawning, and that depends entirely on what kind of a game you want to make. Giving a very high frequency to monsters whose challenge level is close to the dungeon depth gives you more controlled, tunable gameplay where you know exactly what the player is going up against at each depth. Making the choice looser - for instance, you could simply have the depth work as a cut-off such that the probability of spawning any monster beyond that is zero, and have uniform spawn probabilities otherwise - gives breezier gameplay where the deeper the player is, the more they're facing comparatively shallow-leveled monsters (and the more of a relative threat a monster of their actual depth is, of course!).
Then you just tune and twiddle as necessary. Maybe give monsters a base frequency (so that red dragons are more common than turquoise spotted mechano-dracolichs, regardless of which one's a greater threat) . Maybe instead of single depth you could give them a level range, and multiply the base frequency by (say) a thousand if the current level is within the range. Or define spawn probabilities as second-degree polynomials of player depth. Have extra multipliers for themed levels, so that maybe the warrens level has a ten times bigger chance of spawning goblins. Have the super extra dangerous mid-game level have a high chance of spawning very out-of-depth monsters. Etc., etc.. Probabilities are simple and powerful things, you can do a lot of stuff with them once you get used to them.