Cheap-o method : Hit that inside of the cabinet up with some Tilex and let it sit out in the sun for a day or two to kill the mold/mildew. That made a huge difference in ones I've gotten with similar smell/mold. They didn't look quite as funky as this one does though. Then when that is done, clean out the crap, get one of those bags of activated charcoal they sell at lowes and throw that in there with it closed up. After a week or so of that bag being in there, smell is gone unless you put your nose right up to it. Then you just gotta scrub the crap out of it, then paint.
If you can, and since it probably already has moisture damage, an ozone generator will work to remove the smell entirely, at the expensive of extra oxidation. You would be better off partially rebuilding the cab and elbow grease though. At a minimum, replace the bottom wood, then scrub the crap out of the inside with something like simple green.
Springfield, IL
