Crosbie was born in Gorebridge, Midlothian, Scotland, to Presbyterian parents who disapproved of her becoming an actress. Nevertheless, she joined the Bristol Old Vic Theatre School while still in her teens. She began her career with the Glasgow Citizens' Theatre Company in 1956. She was educated at Boroughmuir High School in Edinburgh. Her big break came in 1970 when she was cast as Catherine of Aragon in the BBC television series "The Six Wives of Henry VIII", for which she won the 1971 BAFTA TV Award for Best Actress. In 1973, she starred alongside Vanessa Redgrave in the BBC serial "A Picture of Katherine Mansfield".