It was rare as an American given name before 1975. In the mid-1980s it began growing in popularity for boys after a character by this name began appearing on the soap opera Ryan's Hope. It is now more common as a feminine name, probably due to the fame of the actress Dakota Fanning (1994-).
As a given name it was mostly masculine before 1940, but it rose in popularity as a name for girls during the 40s and 50s, probably due both to Grace Kelly (who married Prince Rainier III of Monaco in 1956) and a female character on the 1957 television series Bachelor Father [1]. By the end of the 1970s it was on the decline.
The name was brought to public attention by Samuel A. Taylor's play Sabrina Fair (1953) and the movie adaptation Sabrina that followed it the next year. This is also the name of a comic book character, Sabrina the Teenage Witch, first introduced 1962 and with television adaptations in 1970-1974 and 1996-2003, both causing minor jumps in popularity. Another jump occurred in 1976, when it was used for a main character on the television series Charlie's Angels.