She was born in Gorebridge, Midlothian, Scotland, to Presbyterian parents who disapproved of her becoming an actress. Nevertheless, she joined the Bristol Old Vic while still in her teens. Her big break came when she was selected to play Catherine of Aragon in the major BBC television series, The Six Wives of Henry VIII in 1971. A few years later, she made a similar impact as another Queen, Victoria, in the ITV period drama, Edward the Seventh (1975).
Crosbie's next major role was as Margaret, long-suffering wife of Victor Meldrew, in the British sitcom, One Foot in the Grave (1990). She also played Janet, the housekeeper to Doctor Finlay, in the 1993 revival of A.J. Cronin's popular stories.