Speaking in Laos, Mr Obama said that every time Mr Trump spoke it became clearer that the Republican contender was not qualified to be president.
In a televised forum on Wednesday, Mr Trump had praised Mr Putin’s “great control” and 82% approval rating.
Mr Trump and rival Hillary Clinton had taken questions from military veterans.
Mr Obama said: “I don’t think the guy’s qualified to be president of the United States and every time he speaks, that opinion is confirmed.”
The president pointed to the diplomatic work he had faced at both the Asean summit in Laos and the earlier G20 meeting in China.
He said: “I can tell you from the interactions I have had over the last eight or nine days with foreign leaders that this is serious business.
“You actually have to know what you are talking about and you actually have to have done your homework. When you speak, it should actually reflect thought-out-policy you can implement.”
Mr Trump had told the forum in New York that the Russian president had “been a leader far more than our president has been”.
Quizzed by NBC host Matt Lauer on his previous complimentary remarks about Mr Putin, Mr Trump responded: “He does have an 82% approval rating.”
“I think when he calls me ‘brilliant’, I’ll take the compliment, OK?” said the businessman, adding that Mr Putin had “great control over his country”.
Mr Trump also said that, as a result of the confidential intelligence briefings he has been entitled to as an election candidate, he had been “shocked” at how the president, Mrs Clinton and current Secretary of State John Kerry had done “exactly the opposite” of what intelligence experts had told them.
In the forum, Mr Trump also said: “I was totally against the war in Iraq.”
This appeared to contradict a statement in a 2002 interview with radio host Howard Stern and the forum’s moderator, Matt Lauer, came in for intense criticism after the event for not pressing Mr Trump on the statement.
Mr Obama said in Laos: “The most important thing for the public and the press is to just listen to what he says and follow up and ask questions to what appear to be either contradictory or uninformed or outright wacky ideas”.
Mrs Clinton had found herself once again on the defensive during the forum over her private email server.
A US naval flight officer told the former secretary of state he would have been jailed if he had handled classified information as she had done.
The Democratic nominee replied: “I did exactly what I should have done and I take it very seriously. Always have, always will.”
The former secretary of state vowed to defeat the Islamic State group, though she emphasised: “We are not putting ground troops into Iraq ever again.”
Mrs Clinton also said her 2002 Senate vote in favour of the Iraq War was “a mistake”.
But she said it meant she was in “the best possible position” to ensure it never happened again.
The forum offered a preview of the questions they will face in their three forthcoming presidential debates, the first at Hofstra University near New York on 26 September.