"cut off the top of a plant," c. 1200, from crop (n.). The general meaning of "to cut off" is mid-15c. Related: Cropped; cropping. Women's fashion crop top is attested from 1984.
"adherent of the Parliamentary party in the English Civil War," 1641, so called for their custom of wearing the hair close-cropped, in contrast to the flowing curls of the cavaliers.