How to properly ventilate a finished basement?
Sounds like dehumidification might be the answer. The cause of the dank "basement" smell, is high humidity. Removing moisture from the air is a side effect of refrigeration, which is why the air conditioner helps.  A dehumidifier is just like an air conditioner, except that the dehumidifier heats the air back up after cooling it and removing moisture.

You may be able to simply use a standalone dehumidifier, though then you'll have to dump the collection bucket. Another option, is to install a whole house (or whole apartment) dehumidifier. A whole house unit will be more expensive, but won't require dumping a bucket or water once a day.



If you have moisture problems in the basement, you'll want to sort those out. That should eliminate the cause of the foul air, and the need for removing it.

If you don't want to deal with the moisture problems, then you'll have to focus on removing the foul air and replacing it.  This means drawing in outside air, conditioning it (heating, cooling, filter, humidify, dehumidify, etc.), and exhausting the foul air.  

If you live in a mild climate where indoor and outdoor temperatures are similar, this shouldn't be that bad. However, if you live where indoor and outdoor temperatures are extremely different, you're going to be paying a lot to condition the air.  

You'll want to look in to a heat recovery ventilation (HRV), and/or energy recovery ventilation system. As these systems will help reduce the amount of heating/cooling that will have to be added to the incoming air.

Drawing air from one dwelling unit into another, is not a proper solution.  Air must be moved within a single dwelling unit, or between the inside and out.