Brady was born in New Orleans but moved with his parents to Bayou Dularge in Terrebonne Parish when he was fourteen years of age.
During World War II, he served in the United States Army Air Forces as an airplane mechanic in the Pacific Theater of Operations.
[2] In 1971, prior to regular two-party competition, he was nominated in the Democratic primary election in 1971 to represent state House District 52, which includes part of Terrebonne Parish.
Brady also worked to adopt United States Coast Guard boater-safety regulations, but many constituents objected to the U.S. government dictating such policy.
Morris Lottinger Jr., a retired judge of the Louisiana First Circuit Court of Appeal who served with Brady in the legislature, described his friend as "the consummate legislator.
[2] As the chairman of the Budget and Finance Committee of Terrebonne General Medical Center in Houma, Brady was one of the few businessmen on the board.