Old English bannan "to summon, command, proclaim," from Proto-Germanic *bannan "to speak publicly" (used in reference to various sorts of proclamations), "command; summon; outlaw, forbid" (source also of Old Frisian bonna "to order, command, proclaim," Old High German bannan "to command or forbid under threat of punishment," German bannen "banish, expel, curse"), apparently a Germanic specialization from a suffixed form of PIE root *bha- (2) "to speak, tell, say" (source also of Old Irish bann "law," Armenian ban "word").
From mid-12c. as "to curse, condemn, pronounce a curse upon;" from late 14c. as "to prohibit;" these senses likely are via the Old Norse cognate banna "to curse, prohibit," and probably in part from Old French banir "to summon, banish" (see banish) and was a borrowing from Germanic. The sense evolution in Germanic was from "speak" to "proclaim a threat" to (in Norse, German, etc.) "to curse, anathematize."
The Germanic root, borrowed in Latin and French, has been productive: banal, 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. Ban the bomb as a slogan of the nuclear disarmament movement is from 1955.