What should I do to this basement closet so I can put clothes in it?
Since you don't mention the problem elsewhere in the basement you can probably solve it with increased ventilation in the closet. You can start by leaving the door open to see if it helps. If you see improvement install some louvers to help circulate some air. You can add a small 4 inch fan to help move more air. If the exposed water pipes are not insulated condensation may be forming on the pipes adding moisture to the air. If opening the closet doesn't help you have two options decrease the humidity or raise the temperature. I spent some time in a tropical area. The closets had a socket for a light bulb typically a 60 watt bulb. With the door shut and the light on the temperature  increases.This allows the air to retain more moisture so mildew won't form. Not the greenest solution though. Another option is a dehumidier either a conventional electric unit or the calcium cloride filled pan depending on how seasonal the problem is.