a complicated word of uncertain history. First attested late 15c. in the sense "lewd person" (of either sex; since c. 1700 applied exclusively to women); probably [Middle English Compendium] from Old French baud "gay, licentious" (from Frankish *bald "bold" or some such Germanic source; see bold), despite the doubts of OED. The 15c. English word perhaps is a shortening of baude-strote "procurer or procuress of prostitutes" (c. 1300).
For the French sense evolution from "bold" to "lewd," compare Old French baudise "ardor, joy, elation, act of boldness, presumption;" baudie "elation, high spirits," fole baudie "bawdry, shamelessness." The Old French word also is the source of French baudet "donkey," in Picardy dialect "loose woman."
The second element in baude-strote would be trot "one who runs errands," or Germanic *strutt (see strut (v.)). There was an Old French baudestrote, baudetrot of the same meaning (13c.), and this may be the direct source of Middle English baude-strote. The obsolete bronstrops "procuress," frequently found in Middleton's comedies, probably is an alteration of baude-strote.