I had this once. When it rained, the moisture came right thru the stucco on the outside surface of the building, and entered
the closet. To cure it, I painted that wall of the outside of the building with an elastomeric paint, available at Sherwin Williams store, and probably Kelly Moore store ($25/gallon).
That kept the moisture from getting in. I wonder if it would work by painting the inside of the closet.
A tenant of my brother in law, he put a lightbulb in the closet. The building caught on fire when the bulb touched some clothes.