mid-13c., "sinfulness, infraction of the laws of God," from Old French crimne "crime, mortal sin" (12c., Modern French crime), from Latin crimen (genitive criminis "charge, indictment, accusation; crime, fault, offense," which probably is from cernere "to decide, to sift" (from PIE root *krei- "to sieve," thus "discriminate, distinguish").
Klein (citing Brugmann) rejects this and suggests *cri-men, which originally would have been "cry of distress" (Tucker also suggests a root in "cry" words and refers to English plaint, plaintiff, etc.). But de Vaan accepts that it is from cernere (compare discriminate).
The meaning "offense punishable by law, act or omission which the law punishes in the name of the state" is from late 14c. The sense of "any great wickedness or wrongdoing" is from 1510s. The Latin word is glossed in Old English by facen, which also meant "deceit, fraud, treachery." Crime wave is attested by 1893, American English.