It's really of deep english origin, often identified as an uninflected form of 'by', but also as a form derived from the same indo-european root as the verb 'be'. This accounts for its numerous meanings (proximate, thoroughly, covered in, on, around, etc., or even when used as just an intensifier for an adjective, verb, or participle)

Mostly like the verb to: be, “am is or are” decked out in, dazzled by etc.

To the best of my knowledge (please correct me if I’m wrong), the prefix ”be-” isn’t related to the verb ”to be” (which comes, BION, from the Old English verb ”beon”—“to be”), but descends from the Old English all-purpose preposition ”bi” (“by”, “near”, “around”, “to”, “in”, “at”, “about”, or “with”). Its lineage goes back at least to the Proto-Indo-European prefix “ambhi-”, and so it’s a close cousin of the prefix ”ambi-” and the preposition ”by”:

The prefix ambi- has the sense of “on both sides” in “ambiguous” and “ambivalent,” while implying “around” in “ambit,” “ambulate” and “amble.” It comes from the Proto-Indo-European prefix *ambhi- “both (sides),” of which the Greek equivalent is amphi-, which we see in “amphibian” and “amphitheater.” In the Germanic languages, *ambhi- dropped its second syllable and became (among other things) the Old Norse um-, a prefix which contributed to . . . the Swedish “ombudsman.” And, taking another route through the same language family, it also lost its first syllable and acquired the sense “beside,” eventually giving rise to the English word “by.”
—“ambisinistrous (2008–05-27)”, “Word of the Day”
http://YourDictionary.com/wotd/ambisinistrous

The modern prefix “be-” also has many functions:

* To intensify the meaning; as, bespatter, bestir.
* To make the action of a verb particular or definite; as, beget (to get as offspring); beset (to set around).
* To render an intransitive verb transitive; as, befall (to fall upon); bespeak (to speak for).
—OneLook Dictionary Search, http://onelook.com/?w=be-

If I had to guess, I’d say it’s a sibling of the German inseparable prefix “be-” (“bemerken”, “bekommen”, usw.). But guessing is an excellent way to be wrong. Does anyone know?

Here are some more references:

The contemporary English prefix “be-”:

Online Etymology Dictionary
http://etymonline.com/?term=be-

Wiktionary
http://en.wiktionary.org/wiki/be-
http://en.wiktionary.org/wiki/Category:English_words_prefixed_with_be-

OneLook Dictionary Search
http://onelook.com/?w=be-

Wordnik
http://wordnik.com/words/be-

The Free Dictionary
http://tfd.com/be-
http://encyclopedia.tfd.com/be-

The Old English preposition “bi”:

Wiktionary
http://en.wiktionary.org/wiki/bi#Old_English

The Latin determiner “ambo”:

Wiktionary
http://en.wiktionary.org/wiki/ambo#Latin

Related contemporary English:

The prefix “ambi-”:

Online Etymology Dictionary
http://etymonline.com/?term=ambi-

Wiktionary
http://en.wiktionary.org/wiki/ambi-#English

The preposition “by”:

http://etymonline.com/?term=by