"stem of a plant," early 14c., probably a diminutive (with -k suffix) of stale "one of the uprights of a ladder, handle, stalk," from Old English stalu "wooden part" (of a tool or instrument), from Proto-Germanic *stalla- (source also of Old English steala "stalk, support," steall "place"), from PIE *stol-no-, suffixed form of *stol-, variant of root *stel- "to put, stand, put in order," with derivatives referring to a standing object or place. Of similar structures in animals from 1826.
"pursue stealthily," Old English -stealcian, as in bestealcian "to steal along, walk warily," from Proto-Germanic *stalkon, frequentative of PIE *stel-, possibly a variant of *ster- (3) "to rob, steal" (see steal (v.)). Compare hark/hear, talk/tell). In another view the Old English word might be by influenced of stalk (n.). Meaning "harass obsessively" first recorded 1991. Related: Stalked; stalking.
A stalking-horse in literal use was a horse draped in trappings and trained to allow a fowler to conceal himself behind it to get within range of the game; figurative sense of "person who participates in a proceeding to disguise its real purpose" is recorded from 1610s.