Turn on the closet light and keep it on. This helps to remove the moisture from the closet, and with it, the moldy and musty smells that accumulate from moisture. The light does not need to be on for 24 hours a day, but an hour a day can work wonders.
Leave the closet door open to air out the closet. Keeping the door closed all the time will trap in odors.
Place a dryer sheet in the closet to keep your sheets and towels smelling fresh.
Place carbonate crystals, activated charcoal or baking soda into a container and place the container in the closet. All three of these materials absorb odors.