Born in Lowell, Massachusetts, Draper attended public, private, and high schools. He studied mechanical engineering and cotton manufacturing. He enlisted as a private in the Twenty-fifth Regiment, Massachusetts Volunteer Infantry, on September 9, 1861 and was promoted through the ranks to lieutenant colonel. He was a brevetted colonel and brigadier general of Volunteers. He became a manufacturer of cotton machinery at Hopedale, Massachusetts, and patented many improvements. He served as delegate to the Republican National Convention in 1876. He served as colonel on the staff of Governor John Davis Long from 1880 to 1883.
Draper was elected as a Republican to the Fifty-third and Fifty-fourth Congresses (March 4, 1893-March 3, 1897). He served as chairman of the Committee on Patents (Fifty-fourth Congress). He was not a candidate for renomination in 1896. He served as president of the Draper Co. upon its incorporation in 1896. Ambassador and Minister Plenipotentiary to Italy 1897-1899. He died in Washington, D.C., on January 28, 1910. He was interred in Village Cemetery, Hopedale, Massachusetts.