The son of a dock worker, he was born in Shadwell, London. His family moved to Hainault in Essex while he was an infant. He attended Hainault High School until the age of sixteen when he left school and took a series of manual jobs. While he was working for London Transport as an underground track-repairer, he started to become involved in union politics. In 1983 he was elected as a local representative to the National Union of Railwaymen (NUR), and then in 1985 he became NUR national officer for track workers.
In 1990 the National Union of Railwaymen merged with the National Union of Seamen to form the National Union of Rail, Maritime and Transport Workers (RMT), and the following year Crow became London Underground representative on the National Executive. In 1991 he became assistant general secretary and then on 14 February 2002 he was elected by the membership to succeed the late Jimmy Knapp as General Secretary. He received 12,051 votes in the election - nearly twice as many as the other two candidates put together (Phil Bialyk received 4,512 votes and Ray Spry-Shute received 1,997).
He has never been a member of the Labour Party. He was a member of the Communist Party of Britain between 1983 and 1997, when he left to join Arthur Scargill's Socialist Labour Party (SLP). He has since left the SLP, and is no longer a member of any political party, but is a supporter of the Socialist Alliance, an umbrella organisation that is attempting to unite the political left in Britain.
He is married, with two daughters and one son.