do the two closets join on a wall??? we just found a leak/mold where the laundry room and a closet shared a wall. If there is no leak then it is just too much humidity.

hmm. sump pump in the basement implies a certain amount of dampness inherent in the house to me...

and if the closet doors are kept closed, you get the combination of damp and stagnant... perfect place for icks to grow.
check the gutters and make sure that they're not doing something evil like draining half the runoff into the eaves and down the exterior wall (very evil, like call the insurance company kind of evil)
if that's not causing the dampness, and
there's no sign of an actual leaking pipe in the wall (actual condensation or staining of the drywall or the ceiling below) then you might just have damp air... slick surfaces like shoes show mildew and mold first, but eventually, it will get into other things, and that's not good.

the simplest answer in a small space is a bucket of 'damp rid'... I actually keep one in my husband's car, since he can literally fog the windows up just sitting there waiting for me to get out of the market!

and making sure that the closet doors get opened on a regular basis.

noodles, china cat,
The two closets are not back to back, no pipes nearby, no stains on walls or ceiling. Gutters are cleand every year, but she is checking to see if there might be a problem there. The sump pump is there because the area is near a river and subject to rising water, but none in recent memory. She saw an ad for damp rid and got that a couple weeks ago and says it is helping. Also keeping the closet doors open more.
Thanks for replying. Every suggestion is appreciated.

Sue

Are the exterior walls filled with insulation? If so, it is possible that moisture is accumulating in the exterior wall, and slowly migrating into the closet. If the problem persists, I think you will have to find someone locally who can inspect the wall, & suggest a solution.

If there are windows near this mildew smell, check to see if the sealant/caulking is intact. It's possible water is seeping in and staying within the walls and the interior insulation is moist amd moldy. Have any baseboards pulled away from the wall?

Both closets are on interior walls. No problems with baseboards.
Right now DampRid is helping a lot.

Thanks
Sue

I have a sump pump that runs every time it rains, but I have no problems with dampness whatsoever.