Born in New Jersey in 1957, Minion briefly attended NYU Film School before finishing his studies at Columbia University, then renowned for its screenwriting program. In 1984, Minion's script for After Hours was optioned by Griffin Dunne and Amy Robinson, the latter a former actress who'd appeared as Harvey Keitel's damaged girlfriend in Scorsese's Mean Streets (1973). Robinson sent Minion's screenplay to Scorsese, whose Last Temptation of Christ had recently fallen through; production on After Hours started shortly afterward. Later, it surfaced that Minion had plagiarized portions of After Hours from radio host Joe Frank, who sued and settled for an undisclosed sum. 
Minion continued his screenwriting career with 1987's Julia and Julia starring Kathleen Turner; 1989's Vampire's Kiss, starring Nicolas Cage; Motorama (1991); and On the Run (1999). He also worked again with Martin Scorsese in 1986 for an episode of Amazing Stories, "Mirror, Mirror".
As a director, Minion made his debut for producer Roger Corman with 1987's Daddy's Boys, said to have been thrown together at the last minute to make use of standing sets for Big Bad Mama II. His last outing as director was for another low-budget feature, Trafficking (1999).
- Joseph Minion at the Internet Movie Database
- The Scandalous Origins of Martin Scorsese’s After Hours