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. Her big break came in 1970 when she was selected to play Catherine of Aragon in the major BBC television series, The Six Wives of Henry VIII.

A few years later, Crosbie made a similar impact as another Queen, Victoria, in the ITV period drama, Edward the Seventh (1975). She played Cinderella's fairy godmother in The Slipper and the Rose, which was chosen as the Royal Film Première for 1976. In that motion picture, Crosbie sang the Sherman Brothers' song, "Suddenly It Happens".

In Ralph Bakshi's animated movie The Lord of the Rings, filmed in 1978, Crosbie voiced the character of Galadriel, Lady of the Elves, and in 1980 she played the abbess in Hawk the Slayer.

Crosbie's next major role was as Margaret Meldrew, the long-suffering wife of Victor Meldrew (played by Richard Wilson), in the British sitcom, One Foot in the Grave (1990 - 2000). She also played Janet, the housekeeper to Dr. Finlay, in the 1993 revival of A. J. Cronin's popular stories.

Crosbie's other roles include playing the monkey-lover Ingrid Strange in an episode of Jonathan Creek, the character of Edith Sparshott in An Unsuitable Job for a Woman, and as Jessie in the 2003 film Calendar Girls.

She also appeared in the sixth series of Old Harry's Game, in which she played a recently deceased historian called Edith. In 2004, Crosbie made a cameo appearance alongside Sam Kelly in an episode of the third series of Black Books, starring as the mother of the character Manny Bianco.