The best solution I've ever had to remove the musty smell from wood furniture is to put it outdoors in the sun, removing all drawers and opening any doors on the furniture so that all parts of the unit are exposed to sun and fresh air. A couple days of this usually works, and the odors disappear.

What a good looking dresser, I love that style and yet I really don't know what it's called. Maybe try putting some cedar shavings in the drawers. Put them in a thin little bag maybe cheese cloth.

The musty smell is probably mildew which needs to be eliminated with laundry bleach. Here are instructions that I found on the internet and used to sucessfully treat an antique dresser with the same issue.

Use a solution of 1 part laundry bleach and 9 parts of water. Be sure that everything in the surrounding area is protected, because the bleach solution will take the color out of carpeting and your clothes and can be rough on many other things. It's best if you can work outside.

You can spray, brush or wipe the solution. The best way is with a pump spray bottle, like window cleaner and such things come in. With the spray you can get in all the nooks and crannies easier, where musty smells and mildew like to hide. Spray until the area you're spraying is damp, but not drippy wet. If you use a brush be sure to get into all the corners and crevices. The same with wiping with a cloth, get the out of the way places real good. You can squeeze the solution from the cloth to get into tight places. Let everything dry naturally.

Be very careful to not get any bleach solution on the finish on the outside of the dresser. â¨â¨Remove all the drawers and apply the bleach solution to all areas of raw wood inside the dresser. Apply to the drawers, inside and out, (don't apply bleach solution to the finish). Mildew can occur on the finish too, but can get deeper into bare wood. If you have some sunshine peeking through in your area, put the and drawers out in the sunshine for a few hours.

After the bleach treatment put the drawers back in and let the dresser sit for a day, then open a drawer and sniff. If the musty odor is still prevalent you'll need to repeat the bleach treatment before going on to the cleaning and orange oil.

Clean the finish on the outside with a good furniture cleaner. Then give the whole thing inside and out, including finish and bare wood, a good application of Orange Oil (see link)

Here is a link that might be useful: Orange Oil

Vodka will remove the odors. It's a very good piece. I would say probably early 1960s. If you don't want it, let me know. I will clean it and buy it some vodka. Bleach will kill any mites and eliminate some of the odors but the reason so many hospitals use alcohol is because if kills bacteria and bacteria is what causes odor. Vodka is a better form of alcohol for furniture (and what the professionals use) than the kind you buy for antiseptic. I also put Listerine (which contains alcohol) in my laundry bins to prevent odors there.