Honestly I can't choose. He has multiple career defining fights, the sign of a true great. The vicious war and knockout of Demarco to annex the title. Or the even more savage rematch Or his revenge win over Saxton, or the epic blitz in the rematch Of course the middleweight win over Robinson is his greatest achievement, although Ray was past his best. And decking Gavilan and taking The Hawk to the absolute brink of defeat and winning in the eyes of some is right up there, given Gavilan's stature, and the fact that he was so much closer to his prime than Ray. And of course there's the brilliant kayo over Malinaggi, which came nearly five decades after Basilio's physical prime.
Of course his wars with Ray Robinson come to mind but people should remember if you subtract those fights with Robinson -- Basilio is still a hall-of-famer.
Yeah he has too many great moments to pick just one, hopefully he will be remembered for All of the great moments.