He made his Newton Heath debut in a 4–0 home win over Darwen on 3 April 1896 and soon became the club's first-choice right-back. By 1897, he had been named club captain, a position he held until he left the club in 1903.
In 1901, Newton Heath was in financial difficulty and held a four-day fundraising bazaar to alleviate the pressure. According to legend, Stafford brought his dog, a St. Bernard named Major, to the event and attached a collection box to its collar with the aim of collecting donations from the public. Major caught the attention of wealthy brewer John Henry Davies, and he and Stafford started to talk about the club and the financial trouble they were in. Davies decided to buy the team and wipe away the debt; he also changed the club's name to Manchester United and their colours to red and white. As a further help to the club, Stafford requested that he be reinstated as an amateur player to take himself off the wage bill.
Stafford made his final appearance for Manchester United in 1903, but remained involved behind the scenes as director, groundsman and scout. In the meantime, he continued to play the odd game for Crewe.
His son, Harry Stafford, Jr., who died on 17 December 1988 at the age of 87, was identified in 2002 as a probable victim of serial killer GP Harold Shipman.