You can try spraying some Febreze in them, cleaning them w/ lemon-scented furniture polish, or taking the drawers outside to "air out" on a sunny day. Cedar cubes/chunks might help some. I have some antique dressers, also... they can have an unpleasant smell!
It depends on what is causing it. If it is that musty smell there is probably some mold. Clean everything with Murphy's Oil Soap. Take the drawers out and wipe them down inside and out and clean the inside of the furniture as well. Once everything is dry use lemon oil polish and wipe everything down again inside and out. That will not only kill any left over mold, but it will moisturize the wood as well.
1) Dehumidify: Wood is porous and therefore will take in moisture, air, and smells of its surroundings. If your piece of furniture has been housed in an old house, attic, basement or garage most likely it has taken in a musty order from one of these. Try moving the piece of furniture into a conditioned room in your home with relatively low moisture in the area. Use a dehumidifier and close the door and leave the piece of furniture for a few hours daily. After a few days, see if this process has helped.
2) Sand & Refinish: Another way to remove musty smell odors is to sand down the original finish and allow the wood to breathe for a few days. Once again, leave the furniture in a conditioned space so it doesn’t absorb new odors and moisture. Choose a stain and refinish the surface with 2-3 coats, allowing the stain to fully dry in between coats. Apply a clear coat finish over the stained furniture to seal in color, and seal out moisture and air.