Crosbie was born in Hankow, China and remained there until 1926 with his family. The Crosbie family moved to Glasgow in 1926 and William attended the Glasgow School of Art from 1932-5. Crosbie gained a scholarship to Paris and there he studied at the Sorbonne and worked under the influence and guidance of Fernand Leger. Exhibiting regularly in Scotland and gained notice in his numerous solo exhibitions. Crosbie is now remembered in the collections of the Scottish National Gallery of Modern Art, Edinburgh, and the National Galleries of Scotland among many others with a retrospective exhibition held at the Scottish Gallery in 1980.