Bassett was born on August 16, 1958 in New York City, New York, the daughter of Betty Jane (née Gilbert; 1935–2014) and Daniel Benjamin Bassett, and was raised in the Harlem neighborhood. Bassett's middle name was given to her in honor of her aunt Evelyn. The origin of the Bassett surname comes from her ancestor William Henry Bassett, who took the surname of his former master, whose children later founded Bassett, Virginia. Ten months after Bassett was born, her mother became pregnant and had a second child, Bassett's sister D'nette. Bassett said the pregnancy "only made things harder." Bassett's parents "shipped" her to stay with her father's sister Golden. While her aunt did not have any children of her own, she "loved children, and she was good with them." After her parents' separation, she relocated from Winston-Salem, North Carolina to St. Petersburg, Florida, where she and her sister D'nette were raised by their social worker/civil servant mother. Bassett did not see her father again for several years, until she attended her grandmother's funeral. There, Bassett met his daughter from his first marriage, Jean, who at twelve years old, was several years older than Bassett. After graduating from Jordan Park Elementary School, she began being bused out of her neighborhood to attend Disston Middle School for seventh grade. The year she began attending was 1970, the first year busing was implemented to integrate public schools in St. Petersburg. After completing seventh grade, she was bused to Azalea Middle School for eighth and ninth grade. Bassett's mother became more involved in her daughter's grades and told her and her sister the pair were going to college.
Bassett first appeared in small roles on The Cosby Show (1984) and Spenser: For Hire (1985), but it was not until 1990 that a spate of television roles brought her notice. Her breakthrough role, though, was playing Tina Turner, whom she had never seen perform before taking the role, in What's Love Got to Do with It (1993). Bassett's performance earned her an Academy Award nomination and a Golded Globe Award for Best Actress.
- Though originally believed that Bassett would be an executive producer for the series, it was later revealed that she would serve as a co-executive producer instead.