I suppose that depends on how you define "responsible for."
As far as how John the Savage dies, he kills himself. Specifically, he hangs himself. So in that sense, he is responsible for his own death as no one helps him die or causes him to die.
If you want to assign moral blame, I suppose you could say the crowd of people (and the society as a whole) is responsible. They are the ones who start chanting "orgy porgy" as John whips himself. They are the one who start the orgy that makes John so disgusted with himself. So you could say that it is they (because of the difference between their values and his) that cause John's death.
John, the Savage, hangs himself at the end of the novel. He had previously asked Mustapha Mond if he could be sent away to the islands with Helmholtz. Mond, however, refused:
“He said he wanted to go on with the experiment. But I’m damned,” the Savage added, with sudden fury, “I’m damned if I’ll go on being experimented with. Not for all the Controllers in the world. l shall go away to-morrow too.”
“But where?” the others asked in unison. The Savage shrugged his shoulders. “Anywhere. I don’t care. So long as I can be alone.”
John doggedly pursues his dream of a life alone, finding a lighthouse and buying seed so he can grow crops. But crowds find him. He whips himself because he believes this will purify him and atone for his sin, and in a society built on the absence of pain, this makes him a media spectacle. He becomes the stuff of newspaper headlines -- and even the subject of a film -- because "pain was a fascinating horror." Newspaper helicopters hover, and hordes of people come to his lighthouse, chanting for him to whip himself.
The implication is that John kills himself because he can find no other way to be alone, no other escape from a society in which he has nothing in common. Who bears responsibility for his death? John bears some responsibility, as he makes the choice. The crowds that won't leave him in peace bear some of the blame, too, though they can hardly be expected to understand what they are doing to John. Perhaps more responsibility lays with Mond, who has both the intelligence and the power to help John but chooses to treat him as an experiment, not a human being.