He lived at Tubrid House, Ardfert, County Kerry.
Crosbie served in the Irish House of Commons as the Member of Parliament for County Kerry between 1759 and 1760.
By his second wife Mary Blennerhassett, daughter of Colonel John Blennerhassett and Jane Denny, he was the father of three children, including John Gustavus Crosbie.
In 1794 he caused a scandal by killing the sitting MP, Sir Barry Denny, 2nd Baronet, in a duel.
His own sudden death three years later was widely rumoured to be a revenge killing by the Dennys.