For the first time since Ronald Reagan was elected, other members of the White House staff are openly criticizing the performance of Michael K. Deaver, the soft-spoken deputy chief of staff who is the president's favorite assistant.

Deaver, an aide since the early days of Reagan's first term as governor in California, is credited with understanding the president's needs and limitations better than anyone in the White House except Nancy Reagan. For this reason, other staffers can't understand why Deaver came up with a European trip schedule that tired the 71-year-old president so badly he nearly blew one of his best media events.

"The wonder of it is not that the president nodded off while listening to the pope," said one official. "It's that he didn't fall fast asleep and really embarrass himself."

Another White House aide observed that Deaver, more than anyone, should have realized that Reagan, who has always liked his sleep, needed more rest than the schedule allowed.

Deaver says that it was the president who made the decision.

"I showed him the schedule and said it was going to be very tough," Deaver said. "He signed off on it. Sure, he gets tired. Who doesn't? But in the long run it's going to be seen as a very successful trip."

Successful or not, the European trip left many strains between the White House and the press corps traveling with the president. Reagan was even less accessible than usual, and so were top White House advisers who took their wives with them to Europe and spent the evenings at social functions. The press was deposited in isolated press centers where reporters were supposed to write the official story without asking troublesome questions.

Resentment of reporters was compounded by haphazard and excessive "security," much of which seemed to have more to do with managing the press corps than protecting the president. Reporters were awakened five or six hours before the day's events so they could be searched before entering press rooms and planes. They were told that the precaution was necessary because someone might have placed bombs in their luggage when no one was looking. But White House staff members whose luggage was similarly unguarded were not searched, leaving the distinct impression that something other than security was in mind.

"It was misplaced security," said NBC correspondent Judy Woodruff, summing up the feelings of many others. "They were protecting the president from people who were no danger to him."

The problems were compounded by an advance operation that seemed overwhelmed by the complex logistical details of moving several hundred people through Europe on a clockwork schedule. "This is my 14th trip abroad with presidents, and it's far and away the most ineptly organized," complained veteran Newsweek correspondent Tom DeFrank.

White House officials have promised to make some changes "next time," which some are betting will be a Reagan trip to the People's Republic of China (not Taiwan) in 1983.

At Templehof Airport, the president launched into his standard pitch about the Berlin Wall, saying that he wanted to ask the Soviets why "they are so afraid of freedom" that they built it. But even on this serious subject Reagan couldn't resist a quip. "In fact, I may stuff the question in a bottle and throw it over the wall when I go there today," he said. . . . Best description of the private meeting between Reagan and Pope John Paul II was by United Press International's Helen Thomas, after the pope assumed a commanding position behind a desk with the visiting president seated respectfully across from him. "You could tell who was asking for the loan," Thomas said. . . . Best dissociation from a previous non-event, by White House counselor Edwin Meese III, when he was asked why the president was the only economic summit participant not to meet the press. "It didn't happen on my watch," said Meese, who replaced White House chief of staff James A. Baker III on the second half of the 10-day trip.

White House strategy is not to make too big a deal about House approval of the "revised recovery budget" last week despite the president's happiness that it passed while he was still in Europe. Some advisers remember the big ballyhoo of 1981, when budget approval was supposed to be followed by a market resurgence based on the prospects of better times ahead. Instead, the market accurately anticipated the growing federal deficit and responded to the "market forces" Reagan sometimes defies, rather than the exhortations of the administration.

This time, the Reagan White House will try hard not to overpromise.

We know now that the markets will want to wait and see whether the spending cuts are actually made," said a White House official.

So while the president will continue to "express pride" in a budget far removed in content from the one he introduced last February, the hope of his advisers is that everyone will be more cautious this time.

White House advisers are very pleased with the results of last week's primaries in New Jersey and California, even though Reagan made a point, as he usually does, of not taking sides in the Republican primaries. But the White House priority for 1982 is keeping control of the Senate and the view among the Reaganites is that this will probably be a lot easier with winners such as New Jersey's Millicent Fenwick and California's Pete Wilson than it would have been with losers such as Jeff Bell (a one-time Reagan speechwriter) or Rep. Barry Goldwater Jr.