To what extent is the protagonist conditioned by his or her physical and social environments?

In the story "Sonny's Blues," the protagonist is, of course, Sonny. Although Sonny's brother takes up the prominent role of narrator, Sonny is still the main figure of interest.
To answer your question, it appears that Sonny is to a great extent conditioned by both his physical and social environments. We learn at the beginning of the story that Sonny has been arrested for dealing heroin. His brother, a teacher (and also the narrator of the story), is disheartened at the news. Meanwhile, an acquaintance of Sonny's meets with the narrator to talk about the situation.
The conversation leaves the narrator feeling even more discouraged. Indeed, the exchange between the narrator and Sonny's acquaintance highlights the reality of the neighborhood's social environment. Many are hopelessly mired in social dysfunction, and it appears that the "solutions" have failed to improve matters.

"Listen. They'll let him out and then it'll just start all over again. That's what I mean."
"You mean—they'll let him out. And then he'll just start working his way back in again. You mean he'll never kick the habit. Is that what you mean?"
"That's right," he said cheerfully. "You see what I mean."

Even the narrator, who is a teacher, still lives in a housing project. He tells us that the housing is new but that it already looks "rundown." He muses that he lives there because it is near where he teaches. Yet, he fears that the prevailing culture of dysfunction remains in his blood and also in Sonny's, who desperately tried to escape from it.
We also learn about the narrator's family life. He relates that Sonny favors their deceased father in many ways. Although their father was a physically imposing man, he deeply valued his privacy. For his part, Sonny mirrors this deep predilection towards discretion and masculine reserve. Despite their outward demeanor of strength, however, both Sonny and his father were emotionally vulnerable men.
We learn that Sonny's father never fully recovered from his brother's death years ago. According to the narrator's mother, his uncle was killed when a car full of drunk white men mowed him down. As for Sonny's father, he was never the same again: "Till the day he died, he weren't sure but that every white man he saw was the man that killed his brother."
Although the story doesn't confirm it, we can speculate that his uncle's murderers were never found, much less held accountable for their actions. This state of affairs greatly affected Sonny's father. According to the narrator, Sonny emotes like their father used to. Ironically, the narrator doesn't realize that he is cut from the same cloth until his daughter (Grace) dies. It takes the narrator much courage before he is able to take up his pen and write to Sonny.
As the story ends, the narrator relates his amazement at watching Sonny perform on a piano in a nightclub (as part of a jazz ensemble). At first, Sonny stumbles on the piano. Then, as he finds his rhythm, he begins to play the music of his life. The melody embraces the lives of everyone Sonny has been connected to. The result is both intriguing and powerful. Through his music, Sonny's message comes through loud and clear: our physical and social environments may greatly influence who we become, but they can never destroy our collective resolve to overcome.