early 14c., "noxious vapor in a coal mine, fire-damp, stifling poisonous gas," 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 (source also of Old High German damph, German Dampf "vapor;" Old Norse dampi "dust"). Sense of "moist air, moisture, humidity" is not easily distinguished from the older sense but is certainly attested by 1706.
late 14c., "to suffocate" (with or as with damp, foul air in a mine), from damp (n.). Figurative meaning "to check or retard the force or action of (the spirits, etc.)" is attested by 1540s. Meaning "to moisten" is recorded from 1670s. Century Dictionary (1897) states that "Dampen is now more common in the literal sense, and is sometimes used in the derived senses." Related: Damped; damping.
1580s, "dazed," from damp (n.). Meaning "slightly wet" is from 1706. Related: Damply; dampness.