Is your basement sealed off completely from the outside air? If not, that would be your source of moisture and musty smell. The solution would be to seal off the basement, run your AC and also run a dehumidifier. It will dry up in a hurry and the musty smell will leave but you may need to clean things up to remove any mold that you can not see. In our basement, it starts to smell musty right away if we do not have the dehumidifier running when it gets muggy out. The cool basement walls and floor will condense out moisture from the air in the same way the cold coils on the dehumidifier collect water. But on days like today in N. WI, the dew point was only about 50 so we opened up all the basement windows to air things out. The dew point is key - 55 to 60 means slightly humid, 60-70 means humid, and 70+ is very humid. If the temperature of your floors or walls are below that dew point, the moisture will collect.
You don't realize how much water is in the air till you empty the gallon of water out of your dehumidifier every few days - and this is just inside air with moisture that makes its way in under doors or through floors (our basement is closed off). If you were to run a dehumidifier with outside air coming in you might collect a gallon or more of water a day!