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