Is continuous exhaust from basement level closet a good idea?
If there is no water source explaining the moisture in the closet, and your basement has a vapor barrier, it is possible that the moisture source is airborne.
The closet may be sufficiently cold and the air in the rest of your house may be sufficiently humid that psychrometrics causes the relative humidity in the closet to be high enough to grow mold.
if the cause is psychrometric...
Installing a continuous fan that exhausts outdoors would help only insofar as the increased airflow is able to maintain a higher closet air temperature than before.  If the makeup air flowing into the closet is warm and dry enough, you might solve the problem.
Perhaps a better solution is to insulate the closet to bring it closer to the temperature in the rest of your home.
You probably should get some wet and dry bulb thermometers and learn how to read a psychrometric chart, then run some basic experiments.  Once you wrap your mind around the psychrometrics at work in your case it should be obvious whether they are the cause, and if so, what you could do about it.