As a young man Crosby was apprenticed as an attorney in Sunderland, but soon moved to London where he practised successfully for several years.
In 1768, already established in City politics, Crosby purchased his return for the corrupt and expensive borough of Honiton. From 1769 he was closely associated with Wilkes (‘one of his most steady partizans’,
In 1774 Crosby unsuccessfully contested London on a comprehensive radical programme. He did not stand at the general election of 1780, but at the by-election of January 1784 he again unsuccessfully contested the City. Probably because of ill-health, he did not stand at the general election. Throughout his life he maintained his interest in City affairs, particularly social and charitable work. His obituary describes him as ‘possessed of an uncommon degree of patience, integrity, and sagacious penetration’, and at his death on 14 Feb. 1793 he was still serving as chairman on the four principal City committees.
