Draper began his career as a product of the youth system at Notts County, making his professional debut in December 1988. A talented midfielder, he became recognized by County supporters as one of the greatest young players ever developed at the club. After attracting the attention of a number of bigger clubs Draper was finally sold during the 1994 close season to Leicester City for £1.25 million.
Draper spent the entire 1994–95 season at Filbert Street, playing 39 times (5 goals) in what proved to be a relegation season. His form attracted the attention of Aston Villa, who paid £3.25 million to keep him in the Premier League. Over the next four seasons Draper became a regular in the Aston Villa starting line-up, helping them to a League Cup win in 1996. Draper fell out of favour in the 1999–2000 season, appearing in only one game to bring his Villa total up to 120 league appearances (7 goals), and was loaned to Rayo Vallecano in Spain. He was sold during that close season to Southampton for £1.5 million. By then approaching veteran stage, Draper added another 24 top flight appearances (with one goal) for the Saints before leaving the game.
He is currently involved in property development.