"a load," Old English byrðen "a load, weight, charge, duty;" also "a child;" from Proto-Germanic *burthinjo- "that which is borne" (cf. Old Norse byrðr, Old Saxon burthinnia, German bürde, Gothic baurþei), from PIE root *bher- (1) "to bear, to carry; give birth" (see infer).
The shift from -th- to -d- took place beginning 12c. (cf. murder). Archaic burthen is occasionally retained for the specific sense of "capacity of a ship." Burden of proof is recorded from 1590s.