Be-

CIDE DICTIONARY

Be-,  [AS. be, and in accented form bī, akin to OS. be and bī, OHG. bi, pi, and pī, MHG. be and bī, G. be and bei, Goth. bi, and perh. Gr. 'amfi` about (cf. AS. beseón to look about). By, Amb-.].

 A prefix, originally the same word as by;  [1913 Webster]
"It is joined with certain substantives, and a few adjectives, to form verbs; as, bedew, befriend, benight, besot; belate (to make late); belittle (to make little). It also occurs in certain nouns, adverbs, and prepositions, often with something of the force of the preposition by, or about; as, belief (believe), behalf, bequest (bequeath); because, before, beneath, beside, between.
In some words the original force of be is obscured or lost; as, in become, begin, behave, behoove, belong."  [1913 Webster]

OXFORD DICTIONARY

Be-, prefix forming verbs:
1 (from transitive verbs) a all over; all round (beset; besmear). b thoroughly, excessively (begrudge; belabour).
2 (from intransitive verbs) expressing transitive action (bemoan; bestride).
3 (from adjectives and nouns) expressing transitive action (befool; befoul).
4 (from nouns) a affect with (befog). b treat as (befriend). c (forming adjectives in -ed) having; covered with (bejewelled; bespectacled).

Etymology

OE be-, weak form of bi BY as in bygone, byword, etc.