The Drapers, originally from Kent, settled at Nether Worton in Oxfordshire, where William became one of the principal landowners.
He had at least one sister Anna, who married Francis Osborne, celebrated as the author of Advice to a Son.
He was appointed by the council to be Captain of the Horse Militia in Oxfordshire on 2 April 1650, and was Governor of Oxford from 9 May 1651.
On 17 September the Council appointed him to guard Wallingford Castle and instructed him to march there "and remain there until you receive further order.
On 30 June 1653 the Council of State ordered "the Committee of Whitehall to provide lodgings there for Sir Charles Wolseley and Mr Draper" and in August he was assigned a stable in the Mews, late lodgings of Mr. Berry and "the stable and coach-house late Mr Simpson's, unless Berry shew cause to the contrary.