Crossman earned a Bachelor of Divinity at Pembroke College, University of Cambridge, and was Prebendary of Bristol. After graduation, he ministered to both an Anglican congregation at All Saints, Sudbury, and to a Puritan congregation simultaneously. Crossman sympathized with the Puritan cause, and attended the 1661 Savoy Conference, which attempted to update the Book of Common Prayer so that both Puritans and Anglicans could use it. The conference failed, and the 1662 Act of Uniformity expelled Crossman along with some 2,000 other Puritan-leaning ministers from the Church of England. He renounced his Puritan affiliations shortly afterward, and was ordained in 1665, becoming a royal chaplain. He received a post at Bristol in 1667, and became Dean of Bristol Cathedral in 1683. He died on 4 February 1683, at Bristol, and lies buried in the south aisle of the cathedral at Bristol.
Several of Crossman's hymns are preserved in the Sacred Harp.