It's important to realize that although the colony's main source of income is its mining operation, the colony will still be doing all sorts of other businesses. Given that this is a space operation, I'd think self-sustenance would have been an original goal, as you don't want your entire mining colony to die out if you lose a food shipment. Thus, your colony should be full of people doing all sorts of jobs, such as making food, recycling trash, and keeping the station in good repair.
Once you achieve self-sustenance, your main problem will be growth. Asteroids may provide metals and minerals, and the sun can provide energy, but for the building blocks of edible carbon-based life-forms (fruits, vegetables, bacon, etc) and water, you're either going to need the ability to transmute elements (unlikely and expensive), or you're going to need to import it.
The question now is if these imports could happen. To that, I'd say it depends on how things are faring on Earth (or whatever the home planet is). Nowadays, if someone asks for a few tons of plant matter in return for a few tons of gold, it's easy to see what the answer's going to be. But in the future, who knows what we'll find useful. Will we be in the midst of a new Dust Bowl, and be in dire need of topsoil and food? Will pollution be a problem, poisioning the water and crops? And what about other colonies? Is the mining colony one of hundreds of other places, all asking for the same commodities? All of these factors are essentially supply and demand, and they will dictate how much the colony will need to mine in order to get what it needs. If Earth is well-fed but in dire need of metals, then your colony will have an easy time getting everything it needs to raise some babies, and even hire on anyone looking for work. As for the overcrowding, it might be the case that the very materials you need to add to the station are the ones you're selling to Earth; thus, you can't make more room because you're still trying to keep everyone fed.
The last thing you need to worry about is when your supply of minerals runs out. Like I said, your station should be built to self-sustain from the start, but after some growth it's anyone's guess if that's still the case. After the American Gold Rush, many boom towns became ghost towns; people just left once the gold ran out. If you don't want that to happen to your station, you've got to keep people there somehow.
