Old English bannan "to summon, command, proclaim," from Proto-Germanic *bannan "proclaim, command, forbid" (cf. Old High German bannan "to command or forbid under threat of punishment," German bannen "banish, expel, curse"), originally "to speak publicly," from PIE root *bha- (2) "to speak" (cf. Old Irish bann "law," Armenian ban "word;" see fame (n.)).
Main modern sense of "to prohibit" (late 14c.) is from Old Norse cognate banna "to curse, prohibit," and probably in part from Old French ban, which meant "outlawry, banishment," among other things (see banal) and was a borrowing from Germanic. The sense evolution in Germanic was from "speak" to "proclaim a threat" to (in Norse, German, etc.) "curse."
The Germanic root, borrowed in Latin and French, has been productive, e.g. banish, bandit, contraband, etc. Related: Banned; banning. Banned in Boston dates from 1920s, in allusion to the excessive zeal and power of that city's Watch and Ward Society.