The Democrat made the promise in a statement where she apologised for suggesting half of the Republican’s supporters were “deplorable” people.
Mr Trump had responded by saying the comment was “insulting” to “millions of amazing, hard working people”.
Mrs Clinton accepted that she had been “grossly generalistic”.
Polls released earlier this week suggest Mr Trump is gaining on Mrs Clinton, and the rivals are neck and neck in the key battleground states of Ohio and Florida.
What she said on Friday
Speaking at a fundraiser in New York, Mrs Clinton called half of Donald Trump’s supporters a “basket of deplorables”.
They were, she told the LGBT event, “racist, sexist, xenophobic, Islamophobic – you name it”.
The rest of the Republican nominee’s supporters, according to Mrs Clinton, were “just desperate for change”.
How the Republicans reacted
In his tweeted response, Mr Trump said, “I think it will cost her at the polls!”
Mr Trump’s running mate, Mike Pence, said: “They are not a basket of anything. They are Americans & they deserve your respect.”
Reince Priebus, head of the Republican National Committee, said Mrs Clinton had shown “her outright contempt for ordinary people”.
Millions of Americans, he said, supported the Republican nominee because they were “sick of corrupt career politicians like Hillary Clinton”.
Other Republicans mocked Mrs Clinton, sharing photos of the crowd in the Florida venue where Mr Trump spoke on Friday.
The hash tag #BasketOfDeplorables has been trending on social media as indignant Republicans attack Mrs Clinton.
“A candidate who writes off half the country as a #BasketOfDeplorables should be disqualified”, was one comment on Twitter.
What Clinton says now
Apologising, Mrs Clinton said: “Last night I was ‘grossly generalistic’ and that’s never a good idea. I regret saying ‘half’ – that was wrong.”
“As I said,” she added, “many of Trump’s supporters are hard-working Americans who just don’t feel like the economy or our political system are working for them.”
But for most of the statement, she attacked her rival, accusing him of building “his campaign largely on prejudice and paranoia” and giving a national platform to “hateful views and voices, including by retweeting fringe bigots with a few dozen followers and spreading their message to 11 million people”.
“David Duke and other white supremacists see him as a champion of their values,” she said in the statement.
It was, she said, “really deplorable” that her Republican opponent was linked to people from the right-wing “alt-right movement”.
Where this is going
Passions are high just over two weeks before the first election debate between Mrs Clinton and Mr Trump on 26 September.
After one of the most bitterly fought election campaigns in living memory, Americans go to the polls on 8 November to elect a successor to President Barack Obama.
The Democrat is standing down after two terms in office.