Trump, leading the polls for the Republican nomination, has issued several controversial remarks about Muslims and Syrian refugees in the wake of last week's attacks in Paris claimed by the Islamic State extremist group.
Clinton took issue with his last night remarks, which appeared to draw comparisons with Nazi Germany.
"This is shocking rhetoric. It should be denounced by all seeking to lead this country," Clinton wrote on Twitter.
Trump was asked by NBC News on Thursday whether he would support creating a database system to track Muslims.
"There should be a lot of systems, beyond databases. We should have a lot of systems, and today you can do it."
As to how he would get Muslims registered, he said it could be done at "different places," not just mosques.
"It would be just good management," he said.
Trump was later asked whether there was a difference between requiring Muslims to register today and Jews registering with Germany's Nazis in the 1930s.
"You tell me," Trump responded.
Fellow Republican candidate Jeb Bush, who has been denounced for suggesting Christian refugees should be prioritized over Muslims, joined Clinton in criticizing Trump for his remarks, saying Americans do not have to abandon their values to be resolute in fighting extremism.
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Identifying ISIS as a major threat to the US, Clinton claimed that she has a plan to defeat this terrorist outfit which now controls a large part of Syria and Iraq.
"We now finally are where we need to be. We have a strategy and a commitment to go after ISIS, which is a danger to us as well as the region," Clinton said during the debate.
"I have a plan that I've put forward to go after ISIS. Not to contain them, but to defeat them," she said.
Sanders and O'Malley agreed with her on threat posed by the ISIS, but differed on the approach to be adopted for this.
Clinton, who was the Secretary of State when US troops stormed a safe hideout in Abbottabad in Pakistan to kill al-Qaeda chief Osama bin Laden, asserted that Americans would go anywhere in the world to attack terrorist groups.
Sanders, the Senator from Vermont, argued that the US needs help from other countries to defeat ISIS.
"I do not believe in unilateral American action. I believe in action in which we put together a strong coalition of forces, major powers and the Muslim nations," he said.
Clinton agreed with Sanders on a coalition against ISIS, saying "I agree with him about that. We've got to build a coalition abroad. We also have to build a coalition at home. Abroad, we need a coalition that is going to take on ISIS."
"They should be Muslim troops. I believe that countries like Saudi Arabia and Qatar have got to step up to the plate, have got to contribute the money that we need, and the troops that we need, to destroy ISIS with American support," he said.
Clinton said the situation with regard to ISIS is different from that of Afghanistan. "We have to support Sunni- Arab and Kurdish forces against ISIS, because I believe it would be not only a strategic mistake for the US to put ground combat troops in, as opposed to special operators, as opposed to trainers, because that is exactly what ISIS wants."
Clinton's remarks were soon criticised by Republican presidential candidates and their party.
"No," tweeted Republican presidential candidate Jeb Bush. "We are not 'where we need to be' in fight against ISIS."
Republican National Committee chairman Reince Priebus alleged that Clinton once again proved inconsistent on the issues and out-of-touch with the American people.
"Clinton's assertion that we are 'where we need to be' on ISIS in the wake of the San Bernardino and Paris attacks shows a complete disconnect with reality and is completely out-of- touch with the American people," he said in a statement.
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