1807, "a driving stroke made with the arms in full swing," perhaps a dialectal variant of sweep (n.), or in part from obsolete swip "a stroke, blow" (c. 1200), from Proto-Germanic *swip-, related to Old English swipu "a stick, whip; chastisement." Other possible sources or influences are Middle English swope "to sweep with broad movements" (in reference to brooms, swords, etc.), from Old English swapan; obsolete swaip "stroke, blow;" or obsolete swape "oar, pole."
1825, "strike with a sweeping motion," from swipe (n.). The slang sense of "steal, pilfer" appeared 1885, American English; earliest use in prison jargon:
The blokes in the next cell, little Charley Ames and the Sheeney Kid, they was hot to try it, and swiped enough shoe-lining out of shop No. 5, where they worked, to make us all breeches to the stripes. [Lippincott's Magazine, vol. xxxv, June 1885]
Meaning "run a credit card" is 1990s. Related: Swiped; swiper; swiping.