Etymology
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bespatter (v.)
"soil by splashing with dirty liquid," 1640s, from be- + spatter (v.). Related: Bespattered; bespattering.
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asperse (v.)
late 15c., "to besprinkle," from Latin aspersus, past participle of aspergere "besprinkle, bespatter" (see aspersion). Meaning "to bespatter someone's character with rumor and false reports" is recorded from 1610s.
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be- 

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).

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