Aides have President Biden take the shorter stairs to board Air Force One. When it comes to news conferences, they yell loudly — and quickly — to end the questions, sometimes stealing a classic awards show tactic and playing loud music to signal the conclusion of the event. And forget about regular interviews with major news publications, including a traditional presidential sit-down on Super Bowl Sunday.

Over the years, some of Mr. Biden’s key aides have gone from letting “Joe be Joe” to wrapping a presidential cocoon around him that is intended to shield him from verbal slips and physical stumbles.

All presidents are shielded by the strictures of the office, yet for Mr. Biden, who at 81 is the oldest person in history to hold the job, the decision is not only situational but strategic, according to several people who are familiar with the dynamic. The cloistered nature of his White House reflects a concern among some of his top aides that Mr. Biden, who has always been prone to gaffes, risks making a mistake.

Those risks were revealed in striking ways during the events that unfolded this week.

After a special counsel’s report on Mr. Biden’s handling of classified documents was published on Thursday, the president was furious with how he was portrayed, viewing the report as a partisan and personal attack that included one of the most gutting experiences of his life — his son Beau’s death.

His aides discussed options, including whether to wait a day to respond. But in the end, the president decided to answer questions from reporters who assembled in a haphazard scrum, rather than a formal news conference.

Aides tried to end the scrum multiple times. But Mr. Biden kept talking, offering a forceful defense of his memory.

He also made mistakes. As he headed toward the door, the president turned back to take a question on the war in Gaza. He criticized Israel’s campaign against Hamas as an “over the top” operation that had led to human suffering in the besieged strip.

He described his work to urge other leaders in the region to to allow humanitarian aid into Gaza. But then he mixed up Mexico and the Middle East while recalling the negotiations.

It was not the only flub.

At campaign events this week, he confused dead European leaders for their living counterparts, saying he had spoken to François Mitterrand, the former French president who died in 1996, and Helmut Kohl, the former German chancellor who died in 2017.

Amid the criticism and concern over his words, some of the people closest to Mr. Biden — including Jill Biden, the first lady — are concerned that the presidency wears on him. A small number of aides close to the first couple keep a scrupulous watch over Mr. Biden’s schedule and hash out the finest details, down to the specifics of a motorcade route.

Mr. Biden has granted fewer interviews and delivered fewer news conferences than any of his predecessors dating back to President Ronald Reagan, which has led to criticism that a president who promised “transparency and truth” at the outset of his term has not done enough to explain his decisions to Americans, particularly on foreign policy.

Even the way Mr. Biden walks to the presidential aircraft is subject to careful management. The president started taking a short flight of stairs directly into the belly of Air Force One, rather than a tall stairway wheeled up to a higher point on the plane, after he tripped and fell over a sandbag during a commencement ceremony this past summer. Now, there is a Secret Service agent positioned at the bottom of the stairs when he disembarks. (Mr. Biden’s immediate predecessor, Donald J. Trump, who is 77, often took the short stairs during bad weather.)

White House officials have not said when Mr. Biden will receive another physical exam. The last one was conducted nearly a year ago by Kevin C. O’Connor, the president’s longtime physician, who declared his patient, then 80, to be “healthy” and “vigorous.”

Outside the White House, Mr. Biden’s allies worry about the optics of his physical appearance, which have become fodder for conservative attacks and online memes. And the issue is not just partisan; a recent poll by NBC News shows that half of Democratic voters say they have concerns about Mr. Biden’s mental and physical health.

His gait is somewhat halting, a characteristic multiple people close to the White House say is partly because of his refusal to wear an orthopedic boot after suffering a hairline fracture in his foot before taking office.

Even so, aides say Mr. Biden will keep increasing the number of appearances that allow him to interact directly with the public, including unscheduled drop-ins at restaurants and shops.

The White House rejected concerns about the president’s mental acuity.

Andrew Bates, a White House spokesman, said in an email that Mr. Biden “is traveling the country at an aggressive rate.” He added that Mr. Biden is using “interviews, speeches and innovative digital events” to deliver his message.

Democrats who have spent time with Mr. Biden in smaller settings, including fund-raisers, private meetings and round tables after events, say he remains sharp — even pugilistic.

Jay Jacobs, the chairman of the New York State Democratic Party, said Mr. Biden spoke without notes at a recent fund-raiser, addressing a range of issues, including foreign policy and the stakes of the election. After the event, the president asked Mr. Jacobs detailed questions about the special election for a House seat in New York’s Third Congressional District.

“The characterization that I’m seeing currently is just unfair,” Mr. Jacobs said. “Yes, his voice can sound older. There’s no question of that. But I will tell you from my personal conversations with him, this guy was on his game.”

Mr. Biden’s allies say that there is no proof that he is unfit for office, and that the coverage of his mistakes — and his age — does not compare with the substance of the things he gets correct.

“I care about the action,” said Robert Wolf, a longtime Democratic donor who was at one of Mr. Biden’s fund-raisers in Manhattan on Wednesday. “I care about the legislation. I care about the people he has around him. I don’t care if he messes up between the Middle East and someone’s name.”

Mr. Wolf said that at the end of a long day of headlining campaign events around New York City, Mr. Biden grabbed a microphone and privately took about a half-dozen questions from a group of donors on Wednesday evening, focusing largely on foreign policy.

Others point to the president’s accomplishments, saying it is time for Democrats to stop attacking him — or harboring quiet hopes for someone to replace him on the ticket — and rally behind his candidacy.

“I am not going to tell voters to not take into account the president’s age. The age of an elected official, and of a candidate for office, is a germane consideration,” said Representative Jake Auchincloss, a Democrat who represents suburban Boston. “But I’m going to encourage them to take into account his full profile and track record, everything he brings to the table.”

Mr. Biden’s allies also say that the president’s legislative accomplishments, from a bipartisan infrastructure bill to a measure intended to increase semiconductor production in the United States, are proof not only of his mental acuity but of his ability to negotiate in pivotal — and unscripted — moments.

“Republicans would have loved to come out of these meetings and say, ‘We’d really like to get something done, but unfortunately, you know, the guy can’t remember anything,’” said Jesse Lee, who worked in communications at the White House’s National Economic Council until November. “It’s not like there’s some sacred cone of silence that, you know, never gets broken except for this.”

Doug Mills contributed reporting.