President Carter, confident that he has dictated the agenda of issues for the last week of the presidential campaign, continued to press Ronald Reagan today on the same subjects that dominated their nationally televised debate Tuesday night.

While aides exuberantly claimed that Carter clearly had made headway in the critical business of winning over undecided voters, the president resumed his assult on Reagan as an "extremely dangerous" opponent of nuclear arms control and nuclear nonproliferation policies.

Answering questions before a friendly "town meeting" audience here, Carter seized on one of the openings offered him Tuesday night by Reagan's denial during the debate that he had ever said that halting the spread of nuclear weapons is "none of our business."

The president cited a newspaper clipping from Feb. 1 in which Reagan was quoted as having said precisely that, and told the audience:

"I've been discussing it, but I'm glad that 80 or 100 million Americans last night could see that Gov. Reagan has another extremely dangerous approach, and that is not concerning the Soviet Union but concerning radical and terrorist nations who don't yet have atomic bombs.

"Every American," he added, "ought to stop and think what will happen to this world if we have no control over nuclear weapons between ourselves and the Soviet Union . . . and if we take the position that it is none of our business if terrorist nations have atomic weapons. That is the single most important issue in this campaign, and I'm glad last night it had a chance to come out."

Carter has tried to make the "war-and-peace" question the central issue from the beginning of the campaign and, in the view of his advisers, he succeeded in keeping it in the forefront before the hugh national audience that watched the debate.

Tonight in Newark, the president showed again how he hopes to exploit some of Reagan's debate statements in the closing days of the campaign. Speaking to a black audience at the Bethany Baptist Church, he recalled Reagan's statement that when he was young "this country didn't even know it had a racial problem."

"Well, Gov. Reagan may not have known it," the president said as the audience shouted its approval, "but to millions and millions of Americans, including some in this congregation, it was simply not a problem, it was a life-long diaster."

While the mood was confident as Carter left Cleveland this morning for his final campaign push, his schedule between now and Election Day, next Tuesday, indicated the political problems that still face him and the uncertainty that continues to hang over the election outcome.

The president barnstormed the Northeast today, moving from Pittsburgh to a rally in Rochester, N.Y., and an appearance in Newark before ending up tonight in Philadelphia.

But his schedule later this week calls for him to spend 2 1/2 precious days trying to nail down his natural geographic base -- the southern and border states of Missouri, Tennessee, South Carolina, Mississippi, Florida and Texas.

The president appeared to be in good spirits as he left Cleveland this morning, and he could only have been cheered by the size and enthusiasm of the crowds he encountered during the day. Several thousand people jammed the street in front of Pittsburgh's aging Trinity Episcopal Church site of the "town meeting."

An even larger crowd greeted the president later today in downtown Rochester. And at both stops, Carter, while hammering away on the "war-and-peace" issue, also called on Democrats to remember their traditional differences with the Republicans.

This will certainly be a constant theme during the last week of the Carter campaign, just as today's large crowds were orchestrated by White House advance teams bent on injecting a sense of winning momentum into the campaign.

At the "town meeting" the president for the first time directly confronted the subject of his 69-year-old opponent's age. The issue was raised by a man who identified himself as a ward constable in this heavily Democratic city and said, "I don't believe that a man that old should be running my country and your country."

"That's an issue that hasn't been raised in the campaign, one that I would not want to raise," replied Carter, who could only have been pleased that the subject came up in such a blunt fashion in the presence of cameras from the three television networks.

The president said Reagan had showed himself during the debate to be "a very strong and very capable campaigner, very sure of himself, very vigorous, and I would not want to insinuate in any way that he was not qualified because of age to be the president. I hope the American people will make a judgment on other issues."