It’s going to depend on how you are defining “damp”. That word is not used in the NEMA enclosure definitions.
For example if if it is “damp” because it is outdoors and getting rained on, that would be 3R. If it is outdoors or indoors subject to being hosed down, that is NEMA 4 or 4X (X if corrosion is an issue). If it is indoors, subject to dripping water or mist from a machine operation, NEMA 12 is fine, NEMA 13 if there is oil in the mist. If “damp” involves occasional submersion, then you need NEMA 6!
If if you are not sure, NEMA 4X covers almost all bases, but is an expensive option. But also if the equipment inside generates heat during use, ie electronics, any sealed enclosure (4, 6, 12, 13) is problematic.