He was born in Shreveport, Louisiana, the son of William N. Blanton (June 23, 1890 – November 27, 1967) and Louise G. Wynn (September 21, 1898 – July 5, 1989).
Blanton was raised in Houston, Texas, where his father was general manager of the chamber of commerce.
She also preceded Blanton in death; he and Virginia "Ginger" Nelson were married on November 30, 2002 (she survived his passing).
He also served on the boards of The Methodist Hospital Healthcare System, the Texas Medical Center, the Houston Zoo, and the Jesse H. Jones Graduate School of Management at Rice University, among others.
During the oil and real estate crises of the 1980s, Blanton served as chairman of the Greater Houston Chamber of Commerce, where he notably helped resolve conflict with effective communication and good working relations.