Cooper was born in Waco, and was the son of Madison Alexander and Martha Dillon (Roane) Cooper. The youngest of three, he had two elder sisters, Lucile and Christine; the latter died the year of his birth. A good student, he chose to attend the University of Texas at Austin, graduating in 1915 with a degree in English; while at the University he was a member of Sigma Alpha Epsilon fraternity. Upon his graduation Cooper returned to Waco to work in the family grocery business, the M. A. Cooper Company, before attending officer's training camp in Leon Springs, and serving as a lieutenant and captain for the U.S. Army in World War I. Returning to Texas after the war, he worked for the family business for a decade before striking out on his own in various business pursuits.