The American held the world welterweight title over four years before stepping up to wrest the middleweight crown from Robinson over 15 brutal rounds in New York in 1957. He was beaten in a rematch six months later.
Basilio, who died in hospital in Rochester, New York, was inducted into the International Boxing Hall of Fame, located in his home town of Canastota, in 1990. The organisation announced its flags would fly at half-mast in his honour.
Hall of Fame executive director Edward Brophy said: "During the 1950s and 1960s Carmen was everyone's hero. They talked about him in the coffee shops, grocery stores, gas stations and barbershops all the time. And they still talk about him today.
"He was loved, respected and idolised. His career and memories will last forever in the Village of Canastota."