He was born in Cuyahoga Falls, Ohio. His father was a contractor who built dozens of homes in Cuyahoga Falls. As a young boy William yearned to fly and loved to build models. He and his buddy Joe Plummer enjoyed walking to Stow Airport to watch airplanes take off and land. When mechanics worked on flying machines, William pestered them with questions. A friendly pilot gave Billy his first ride on a biplane when he was 7 years old, he was enthralled, enjoying the world from a different perspective, but slightly baffled that it "didn't seem as fast as he thought".^{[citation needed]} He graduated from Central High School in Washington, D.C. in 1939.

He obtained his CAA private and commercial licenses in 1940, and in 1941 received his instructor's rating at the Wilmington, Delaware airport. In November 1942 he joined the United States Army Air Forces and was commissioned a 2nd Lieutenant. He was assigned to Air Transport Command North Atlantic Wing ferrying B-17 Flying Fortress bombers to England and North Africa. He later was assigned to the Air Transport Command India-China Division "Fireball Line," which flew supplies and equipment in the China-Burma-India Theater from Miami to Assam, India. After the war William became a pilot for the Special Air Mission Squadron at National Airport in Washington.

General Dwight D. Eisenhower, needed a new pilot and sought a trustworthy, no-nonsense aviator to serve with him. William, a straight arrow who did not drink or smoke and was fastidious about being on time, won the job. From 1950 to 1952 he was the personal pilot for General Eisenhower, who was the Supreme Commander of the Allied Powers. Eisenhower personally asked William to serve as his personal pilot and Air Force Aide after becoming president. Colonel Draper was the pilot of Eisenhower's presidential plane "Columbine II," the predecessor of what we now call "Air Force One." His crew was on call 24 hours a day to fly the president anywhere in the world, with his Maryland home having a direct line to the White House. Draper remained as the President's pilot during the next eight years, during which he piloted the President on nearly all his trips and logged over 300,000 miles. Besides the president and First Lady, Draper flew Vice President Richard Nixon, Secretary of State John Foster Dulles and other members of the Cabinet. His passenger list included British Prime Minister Winston Churchill, Soviet Union leader Nikita Khrushchev, Saudi Arabia's King Saud, Greece's King Paul and Queen Frederika, Denmark's King Frederick IX, Japan's Crown Prince Naruhito, Indonesia's President Sukarno and Thailand's King Bhumibol. He also appeared on an episode of What's My Line on May 22, 1955.^[2] There were rare mishaps and malfunctions, but Draper kept his cool aboard Columbine II and its successor, Columbine III. In a thick fog in Georgia, he was forced to land the plane 50 miles earlier than planned. On a return flight from Buenos Aires, the plane developed engine trouble and landed in Surinam. The best-known incident was in 1953 as Draper flew Eisenhower to Florida. Air traffic controllers briefly confused the Columbine II, designated as Air Force 610, with Eastern Airlines Flight 610. Draper suggested a new designation, Air Force One, to avoid future problems. The name stuck and is still in use to this day.

After being assigned to the Alaska Command when Eisenhower left the White House in January 1961, Draper suffered a heart attack in 1963 which ended his service as a pilot, which he had been continuously since he was 19.^[3] William yearned to fly, that's all he wanted to do. His wife said he became despondent when he lost his career. On November 26, 1964, he committed suicide by hanging himself,^[3] reportedly depressed. Draper was interred at Arlington National Cemetery, Virginia.