From the Oxford English Dictionary:
Forms: ME crym, ME–16 cryme, ME– crime; Sc. pre-17 chryme, pre-17 17 cryme, pre-17 17– crime.
Etymology: < Anglo-Norman and Old French, Middle French crime (French crime ) sin, wrongdoing, criminal act (12th cent.; in Middle French also accusation (15th cent.)) < classical Latin crīmen charge, accusation, matter for accusation or blame, reproach, offence, misdeed, in post-classical Latin also sin (late 2nd or early 3rd cent. in Tertullian) < the base of cernere cern v.1 + -men (see -ment suffix). Compare Old Occitan crim, Catalan crim (13th cent.), Spanish crimen (13th cent.), †crim (14th cent.), Portuguese crime (13th cent.), Italian crimine (a1306).