"a load, that which is borne or carried," Old English byrðen "a load, weight, charge, duty;" also "a child;" from Proto-Germanic *burthinjo- "that which is borne" (source also of Old Norse byrðr, Old Saxon burthinnia, German bürde, Gothic baurþei), from PIE root *bher- (1) "to carry," also "to bear children."
The shift from -th- to -d- began early 12c. (compare murder (n.), rudder, afford). Archaic burthen is occasionally retained for the specific sense of "capacity of a ship." Beast of burden is from 1740. Burden of proof (Latin onus probandi) "obligation on one party in an action to establish an alleged fact by proof" is recorded from 1590s.