Anthony Nesty was born in Trinidad in 1967, the youngest of five children in his family. Nesty's family migrated to Suriname when he was seven months old, and he started swimming at the age of 5. Nesty trained and competed in Suriname and the Caribbean through the beginning of his teenage years. He represented Suriname along with his sister, Pauline, at the 1983 Pan American Games. After placing twenty-first in the 100-metre butterfly at the 1984 Summer Olympics in Los Angeles, Nesty enrolled in The Bolles School in Jacksonville, Florida, a prep school with an athletic program known for training elite, world-class swimmers. While training under Bolles coach Gregg Troy, Nesty broke the prep school 100-yard butterfly record held by Pablo Morales. Breaking Morales's record was the beginning of prominent successes for Nesty. He graduated from the Bolles School in 1987.
Nesty returned to international competition at the 1987 Pan-American Games in Indianapolis, Indiana, winning a gold medal in the 100-metre butterfly and a bronze medal in the 200-metre butterfly.
At the 1988 Summer Olympics in Seoul, Korea, Nesty edged American favorite Matt Biondi by one one-hundredth of a second to win the 100-metre butterfly; he finished the event in 53.00 seconds and Biondi in 53.01. Nesty is the only Olympic medal winner from Suriname and after winning his Olympic gold medal, he was unbeaten in the 100-metre butterfly event for three years. Nesty was the second black athlete to win an individual Olympic medal in swimming following Enith Brigitha at the 1976 Summer Olympics, and only the second South American swimmer to win an Olympic gold medal after Alberto Zorrilla in the 1928 Summer Olympics.
Nesty's victory in Seoul was a momentous social and political event for Afro-Caribbeans. The Suriname government commemorated his gold-medal performance on a stamp and on gold and silver coins. A 25-guilders bank note portraying an illustration of a butterfly swimmer was printed in his honor. Surinam Airways named one of its planes after Nesty, and the indoor stadium in Paramaribo was renamed for him.
Nesty won gold medals in the 100-metre butterfly at the Goodwill Games in 1990 and the FINA World Aquatics Championships in 1991. At the 1991 Pan-American Games in Havana, Cuba, he again won a gold medal in 100-metre butterfly and a silver in the 200-metre butterfly. He attempted to defend his 1988 Olympic gold medal in the 100-metre butterfly at the 1992 Summer Olympics in Barcelona, Spain, but finished with a third-place bronze.
After winning his gold medal in Seoul, Nesty accepted an athletic scholarship to attend the University of Florida in Gainesville, Florida, where he enjoyed a successful swimming career with the Florida Gators swimming and diving team under coach Randy Reese and coach Skip Foster from 1989 to 1992. During his four years of National Collegiate Athletic Association (NCAA) competition, he won three consecutive NCAA individual championships in the 100-yard butterfly (1990, 1991, 1992), one in the 200-yard butterfly (1990), and one as a member of the team's 400-yard medley relay team (1991), and received sixteen All-American honors. Nesty also won eleven Southeastern Conference (SEC) titles—five in individual races and six as a member of Gators relay teams.
Nesty graduated from the University of Florida with his bachelor's degree in 1994.
Nesty was inducted into the International Swimming Hall of Fame (ISHOF) as an "Honor Swimmer" in 1998, and the University of Florida Athletic Hall of Fame as a "Gator Great" in 2002.
In the mid 1990s, Anthony returned to the Bolles School as a member of the swim coaching staff.
Currently, he is the associate head coach for the Florida Gators men's swimming team.