1550s, "to strike, punch," which is perhaps from Middle Dutch dossen "beat forcefully" or a similar Low German word. Meaning "to strike or lower (a sail) in haste" is recorded from 1620s; that of "to extinguish (a light)" is slang from 1785; perhaps influenced by dout (1520s), an obsolete contraction of do out (compare doff, don).
OED and Century Dictionary regard the meaning "to thrust or plunge into water, to throw water over" (c. 1600) as a separate word, of unknown origin (perhaps Scandinavian), though admitting there may be a connection of some sort. Wedgwood thinks this word imitative of the sound of the action. Related: Doused; dousing.