Curtis was born in Dartford, Kent, and grew up in Platt. His father was Martin Curtis (10 August 1917 – January 2002), a Cinematographer from Sevenoaks who worked with Humphrey Jennings. His family had a left wing background. Curtis attended the Sevenoaks School on a county scholarship. He completed a Bachelor of Arts degree in human sciences at Mansfield College, Oxford. He started a PhD, during which he taught in politics, but he became disillusioned with academia and decided to leave.