late 14c., "to suffocate," from damp (n.). Figurative meaning "to deaden (the spirits, etc.)" attested by 1540s. Meaning "to moisten" is recorded from 1670s. Related: Damped; damping.
early 14c., "a noxious vapor," perhaps in Old English but there is no record of it. If not, probably from Middle Low German damp; ultimately in either case from Proto-Germanic *dampaz (cf. Old High German damph, German Dampf "vapor;" Old Norse dampi "dust"). Sense of "moisture, humidity" is first certainly attested 1706.