Great ideas, everyone! My room isn't in a basement or subject to moisture. However, the fabrics likely don't get much, if any, air in the closet's cloth/cardboard bins, the dresser or the plastic storage caddy. I'm intrigued by the odor eater inserts and a few bars of soap (have some travel size). And, thanks for the warning about cedar balls possibly staining fabrics. I am sensitive to the smell of lavender and to many room fresheners. Keeping it as simple as possible without adding much fragrance is best for me. Plus, if I make something for someone else, I don't want it to come with a fragrance they may not like.
Funny, the baking soda idea is probably the easiest but I limit using it to the refrigerator. I just may try that in the closet itself and then add soaps or charcoal inserts to individual containers/drawers where they might be most helpful.
Good project for Labor Day weekend....freshening up the fabrics!