Cooper was born on May 7, 1901, in Helena, Montana to Alice (née Brazier, 1873–1967) and Charles Henry Cooper (1865–1946). His father had emigrated from Houghton Regis, Bedfordshire and was a prominent lawyer, rancher, and (later) a Montana Supreme Court justice. His mother had emigrated from Gillingham, Kent and married Charles in Montana.
In 1906, Charles purchased the 600-acre (240 ha) Seven-Bar-Nine cattle ranch about fifty miles (eighty kilometers) north of Helena near the town of Craig on the Missouri River. Frank and his older brother Arthur spent their summers there and learned to ride horses, hunt, and fish.
When Cooper was fifteen his hip was injured in a car accident. On his doctor's recommendation he returned to the Seven-Bar-Nine ranch to recuperate by horseback riding. The misguided therapy left him with his characteristic stiff, off-balanced walk and slightly angled riding style.