Arch's father was a traveling salesman from Ireland and his mother was from a wealthy Canadian family of lawyers. He was rebellious as a teenager and was sent to a boarding school at one point. He got an architecture degree from the Armour Institute of Technology in Chicago in 1915, not on the dime of his parents, but on the dime of a doctor from Omaha, Albert Condon, who was patronizing his future wife also. Arch would later design the doctor's mansion, which still stands. He also designed the swimming pool at his hospital and another hospital in Grand Island, Neb. Finally, he managed Nicholas Senn hospital after Condon's death. In thanks for his patronage, Arch and Mary named one of their two sons Paul Condon Griffith.

Arch was a person known for living life to the fullest. He was relatively tall and of husky build, and his look was very stereotypically Irish in old age. Sweater, vest, spectacles, white hair swooped to the side, the whole bit. He, like his wife, loved to travel. In fact, in later years (after an unfortunate lawsuit), the two lived in an RV in Arizona during winters. He read ancient philosophers voraciously and was an ardent democrat, a fact probably related to his longtime employment as a government engineer. Arch was a Mason and could best be described as an agnostic religiously. He smoked, drank, gambled on horse and dog races, and ate steak every night. Not surprisingly, he eventually had a heart attack, which happened about a month before his actual death. He and his wife are buried less than 200 feet from his parents. They raised their four children in a house he designed himself. It was at 60th and Leavenworth, a brand new development at the time, and Arch designed several houses on the block.

In a 2006 oral history, his daughter Adrienne speaks of him as someone she worshipped, declaring him to be of an extremely high intellect and morality, and the type of person who commands respect.