Just like in the refrigerator, try using a box of baking soda inside of the closet to absorb odors.
Baking Soda Mopping Solutions
If your closet has a musty smell, try mopping an uncarpeted floor with a solution of one gallon of warm water, 1/2 cup of vinegar, and 1/4 cup of baking soda.
Baking Soda and Shoes
Try sprinkling baking soda into your shoes and/or directly on the closet floor to absorb odors.
Do this on a monthly or weekly basis.
Baking Soda and Hampers
Try sprinkling baking soda directly in the bottom of your hamper, or create baking soda sachets to place in your hampers.