late 14c., basken "to wallow" (especially in warm water or blood, of unknown etymology. The Middle English Compendium rejects the derivation from Old Norse baðask "to bathe oneself" (with loss of middle syllable), reflexive of baða "bathe" (see bathe) + Proto-Germanic *-sik "one's self" (source also of German sich; see -sk).
Meaning "soak up a flood of warmth" is apparently due to Shakespeare's use of the word in reference to sunshine in "As You Like It" (1600). Related: Basked; basking.