don't let damp clothes sit in the hamper.
this is definitely a dampness problem. where there is regular dampness, like florida or new jersey, there is mildew or mold. you cab sprinkle cornstarcth or any other absorbent stuff. try those little packets of granular beads that the put in shoes, in your hamper.
the main thing is to get the clothes out of there as soon as possible. i had that problem and the only thing i could do was run a dehumidifier day and night. uh uh! now i open a teeny little window and leave it open (only a monkey can sneak in) not in the winter - and when do a wash, i run a fan. i find that in the winter when the heat is one, it's much better. is that true with you, too? that's because the dampness is less and so is the mold and mildew.
you can see from the answers that it's a common problem with closed-in washers and dryers. quiltncrazy and i are supporting the vinegar industry.