word-forming element of verbs and nouns from verbs, with a wide range of meaning: "about, around; thoroughly, completely; to make, cause, seem; to provide with; at, on, to, for;" from Old English be- "about, around, on all sides" (the unstressed form of bi "by;" see by (prep.)). The form has remained by- in stressed positions and in some more modern formations (bylaw, bygones, bystander).
The Old English prefix also was used to make transitive verbs and as a privative prefix (as in behead). The sense "on all sides, all about" naturally grew to include intensive uses (as in bespatter "spatter about," therefore "spatter very much," besprinkle, etc.). Be- also can be causative, or have just about any sense required. The prefix was productive 16c.-17c. in forming useful words, many of which have not survived, such as bethwack "to thrash soundly" (1550s) and betongue "to assail in speech, to scold" (1630s).
late 15c., "to discolor, to make dirty," of uncertain origin, perhaps from Old French esmorcher "to torture," perhaps also "befoul, stain," from es- "out" (see ex-) + morcher "to bite," from Latin morsus, past participle of mordēre "to bite" (see mordant). Sense perhaps influenced by smear. Sense of "dishonor, disgrace, discredit" first attested 1820.
<a href="https://www.etymonline.com/word/besmirch">Etymology of besmirch by etymonline</a>
Harper, D. (n.d.). Etymology of besmirch. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/besmirch