Ali, who died Saturday (AEST) at age 74, was recalled for his sacrifices and struggles for social justice, as well as his boxing career, by the two giants of the NBA.
The late boxer lost three prime years of his career for his stand against fighting in the Vietnam War before the US Supreme Court vindicated him in 1971.
"When an icon like Muhammad Ali passes away, it's just very emotional," James said.
"It's also gratifying to know that guy, one man, will sacrifice so much of his life knowing it will better the next generation of men and women after him."
James said he would not be among the world's richest and best known athletes if not for Ali.
"I wouldn't be able to walk in restaurants, go anywhere where blacks weren't allowed back in those days," he said.
"Today I can make trips to China and all over the world and people know my name and face. I give all credit to Muhammad Ali because he was the first icon. He's the greatest of all time and it has nothing to do with his accomplishments in the ring.
"His legacy will obviously live on."
Curry said Ali offered "an example of how you use your platform and speak what you believe no matter what people will say".
Warriors coach Steve Kerr began practice with an Ali tribute song including the lyrics: "I'm Ali, the black Superman. I'm Ali, catch me if you can."
"What he did in terms of trying to promote equality, raising the level of consciousness about what was happening in this country - [he was] probably the most influential athlete in the history of our country," Kerr said.