Born in Hartford, Connecticut, Gaffey graduated from Worcester Academy in 1916 and later attended Officers Training School at Fort Niagara, New York, and was commissioned as a second lieutenant in the Field Artillery Branch of the United States Army Reserve on August 15, 1917, four months after the American entry into World War I. Assigned to the 312th Field Artillery Regiment, part of the 79th Division, at Fort Meade, Maryland, in August 1918 he was sent to the Western Front to serve with the American Expeditionary Force (AEF). He later served in Germany with the 12th Field Artillery Regiment, part of the 2nd Division, before returning to the United States in August 1919, nine months after the war came to an end with the signing of the Armistice with Germany.