This use of “be-” intensifies the verb; I am pleased to note that in this use, “befall” takes an indirect object, that least visible of Modern English cases.  This use of befall is not archaic although others (which have more to do with inheritances or actual objects on one’s head) are listed as obsolete!

• 08.129 They wondered what evil fate had befallen him,

Fate doesn’t just fall on someone, but crashes all down around; here the dwarves wonder about Thorin when he is separated from them In Mirkwood.

“befall, v.” OED Online. Oxford University Press, March 2015. Web. 7 May 2015.