Beckham was the son of Richard Olen and Maude Knight Beckham. He grew up in Birmingham and graduated Phillips High School. He attended Birmingham-Southern College, but left to join the U.S. Army Air Corps for service in World War II. While in the service he was commissioned a 1st Lieutenant and assigned to pilot a B-24 bomber in the 8th Air Force. Beckham flew 36 missions in the European theater, including two D-Day bombing runs. He was awarded the Distinguished Flying Cross, the Air Medal with three Oak Leaf Clusters, and four Bronze Stars. Beckham discussed his wartime experiences on the Alabama Public Television program "WWII: Alabama Remembers" in 2007.
After returning home, Beckham married Josephine Milton and settled in Talladega for a while before returning to her home town of Bessemer. Beckham went to work with Alabama Power as an assistant to president Walter Bouldin. He later took over management of the communications and training department and advanced to direct the company's educational and institutional affairs.
Beckham chaired the Alabama Advisory Council on Vocational Education and was a president of the Bessemer Area Chamber of Commerce. He served on the Alumni Advisory Board of the Auburn University School of Business, the Birmingham Area Private Industry Council, and the Girls Club of Bessemer. He also served on the board of stewards and the chancel choir of Bessemer First United Methodist Church.
In retirement, Beckham moved into Greenbriar at the Altamont before leaving to live in hospice care during his final days. He died in August 2013 and is buried at Cedar Hill Cemetery in Bessemer. He was survived by three children, four grandchildren and three great-grandchildren.