Brashear was born on January 19, 1931, in Tonieville, Kentucky, the sixth of eight children to sharecroppers McDonald and Gonzella Brashear. In 1935, the family settled on a farm in Sonora, Kentucky. Brashear attended Sonora Grade School from 1937 to 1946.
Brashear enlisted in the United States Navy on February 25, 1948, shortly after the Navy had desegregated. He graduated from the U.S. Navy Diving & Salvage School in 1954, becoming the first African-American to attend and graduate from the Diving & Salvage School and the first African-American U.S. Navy Diver. Brashear was also the first African-American U.S. Navy Master Diver and the first amputee diver to be certified or re-certified as a U.S. Navy diver.