Joseph was born to J. W. and Christine Braswell in Sheffield and grew up there, moving to Homewood while he was a student at Birmingham-Southern College, where he graduated in 1947. While in Birmingham he designed sets for the Starlight Opera and did interior design for friends and friends of friends.
He then moved to New York City to study at the Parsons School of Design, graduating and teaching and eventually becoming Dean. He left academia with a long list of notable clients that included King Fahd of Saudi Arabia (for whom he outfitted a private airplane) and Greek shipping tycoon John Latsis (for whom he did interiors for several houses).
After September 2001, Braswell moved to Pensacola, Florida. In 2006 he was diagnosed with pancreatic cancer and died on June 16 at the age of 78. His younger sister, Margaret Braswell Gregg, succumbed to cancer on the same day.
- Carlton, Bob (August 27, 2006) "A life lived well, by design." Birmingham News.