Born in Hankow, China, to Scottish parents, Crosbie moved to Glasgow in 1926. He studied at Glasgow School of Art from 1932 to 1934, after which time he was able to travel in Europe on a Haldane Travelling Scholarship. From 1937 to 1939 Crosbie lived in Paris, where he studied under Léger and Maillol. In the 1940s and 1950s Crosbie made his name as a mural painter. His commissions included a mural for the Festival of Britain in 1951. Crosbie also produced illustrations for books and designed the scenery for a ballet.