Mieuli was 89 and died of natural causes after recently being hospitalized, the San Francisco Chronicle reported Monday.
As the owner of the Warriors for 24 years, Mieuli was known for his untamed beard and casual dress, most often topped with a deerstalker cap, and was beloved by Warriors fans, the Chronicle said.
Mieuli also had partial stakes in the San Francisco 49ers and Giants, and operated a radio and television production company in the Bay Area for more than 60 years.
In 1975, the Warriors won the NBA championship with a four-game sweep of the Washington Bullets.
"Franklin was truly one of the innovators in our league," NBA Commissioner David Stern said, "who was so proud of the Bay Area and his ability to maintain a team there."
Mieuli is survived by his longtime companion, Blake Green, son Peter Mieuli and daughter Holly Buchanan, and seven grandchildren, the Chronicle reported.