Mincy was born to Americans Shelly and Willie Mincy at Ramey Air Force Base in Puerto Rico. As an 18-year-old he tried to make the roster of the NBA's New York Knicks. He was unsuccessful and signed instead with the Vaqueros de Bayamón in the National Superior Basketball League of Puerto Rico (BSN).
The same year he accepted a scholarship to UAB, where he helped lead the team to four consecutive NCAA tournament appearances and to 50 wins in his last two seasons. In four years he scored 1,473 points (10.8 points per game average, 8th all-time at UAB), pulled down 933 rebounds (a UAB record) and had a career field goal percentage of .477 (.638 from the free-throw line). He was twice named to the All-Sun Belt Conference first team and was an honorable mention in the All-American voting for 1986.
After college, the Knicks picked him in the 5th round (94th overall) of the 1986 NBA draft, but he did not make the team's active roster. He returned to Puerto Rico and continued to play for the Vacqueros as well as with the Puerto Rican National Basketball Team, which he had joined in 1983. He led the national team in four Olympic Games as well as other international competitions, including a landmark defeat of the United States national team in the 1992 Pan-American Games.
Mincy has been linked romantically to Puerto Rican model and television host La Taína. His younger siblings Jada and Jordan have both played college basketball. Jerome Mincy bobblehead dolls were given away at UAB's March 5, 2004 game against Texas Christian University.
Mincy announced his retirement after the 2002 season, which ended with the Leones de Ponce beating Bayamón in 7 games for the BSN championship. He subsequently signed a one year contract with the Leones, even while the Vacqueros were retiring his #8 jersey.