To get a musty smell out of your closet remove all the clothing and anything else in the closet.
Then sweep or vacuum the closet out really good and then mop the floor if it's not carpet and if carpet use a carpet cleaner with some vinegar and water on the carpet in the closet to remove the musty smell.
Wipe down the walls too with a damp cloth of vinegar and water to help remove the musty smell.
Then put a open box of baking soda inside the closet which can help absorb the musty odor and make sure you don't overfill your closet because a lack of airflow in the closet can cause the musty smell to come back.