Kaye was born Nora Koreff in Brooklyn, New York to emigrant parents from Tsarist Russia, but later Americanized her surname. In 1936, she joined the American Ballet, directed by George Balanchine. She later became a member of the Radio City Music Hall corps de ballet and danced in several Broadway productions, including Giselle (1941), Antony Tudor's Pillar of Fire (1946), and Two's Company, a 1952 revue starring Bette Davis. She worked as an assistant on I Can Get It for You Wholesale (1962), Tovarich (1963), and On a Clear Day You Can See Forever (1965).
Kaye's November 1948 marriage to Isaac Stern ended in divorce the following year. She married Herbert Ross in August 1959. The couple collaborated on several screen projects, including Goodbye, Mr. Chips (1969), The Last of Sheila (1973), Funny Lady and The Sunshine Boys (both 1975), and The Seven-Per-Cent Solution (1976).