President Reagan, sounding like a candidate running against big government, said yesterday that the private sector holds the key to the nation's future.

He also predicted that the recession will end "faster than expected" because his economic program "is based on common sense: reducing the percentage of GNP taken by government."

On his way here, Reagan took a short detour to look at the Potomac River near the 14th Street Bridge where an Air Florida jetliner crashed Thursday. The president ordered the Marine Corps helicopter flying him from the White House to Andrews Air Force Base to pass over the crash site.

Reagan was "saddened and concerned" by his glimpse of the disaster scene and of recovery operations, according to assistant White House press secretary Mark Weinberg. The president indicated he may telephone some of the survivors.

Reagan struck a familiar anti-big-government, pro-private-initiative theme and mixed in a gibe at the Soviet Union in his speech to a luncheon of New York Partnership, Inc., a group dominated by businessmen and headed by David Rockefeller, the former chairman of Chase Manhattan Bank.

"You in the private sector--corporations, firms, merchants, family farmers, Mom and Pop stores all over the country--you hold the key," Reagan said.

He added that he is confident his economic program will work because "it places a premium on individual initiative."

Reagan lavished praise on New York Partnership as an example of the kind of private initiative that he says is essential to economic and social progress. "You are that tough, little tug that can pull our ship of state off the shoals and out into open water," he said.

New York Partnership, organized in 1979, created more than 13,000 summer jobs in private businesses for disadvantaged youths last year. "Nothing that large or successful has ever been achieved anywhere else," Reagan said.

In contrast, he said that government "social experiments" were well-intentioned, "but too often those meant to benefit most from government-imposed solutions paid the highest price and bore the deepest scars when they failed." He equated big government with a loss of freedom: "The brutal, Soviet-sponsored repression in Poland reminds us how precious our blessings are. Ironically, the Soviets understand and agree. They believe freedom is precious, too--they ration it like all the other good things their people don't have."

Reagan said, "This can be an era of losing freedom or one of reclaiming it. I think we have made our choice and turned an historic corner. We are not going back to the glory days of big government."

In urging greater private-sector charitable work, Reagan quoted the Rev. Billy Graham as estimating that if the nation's churches and synagogues adopted an average of 10 poor families each "we could eliminate all government welfare in this country."

But Reagan said he is not asking the private sector to replace--dollar for dollar--every cut in the federal budget. The same benefits can be achieved at less cost, he said. "Private human capital is far more valuable and effective than federal money."

The president cited successful substitution of private initiatives for government actions in Arizona, Boston and Washington, D.C. Jubilee Housing in Washington has found jobs for 55 welfare recipients and claims it can renovate housing far cheaper than federally financed programs do, he said.

Reagan, as he frequently does in speeches, quoted from letters sent to him by Americans who support his economic program.

A blind 94-year-old woman in Odessa, Fla., wrote to say that she was "thrilled" at his efforts to make America "a nation of neighbors who care about each other; families who take care of their own," he said.

A Massachusetts man in his 80s, Reagan said, sent his monthly Social Security check "to be used for reducing the national debt."

Reagan even described a letter he has not received. No one has written him asking "will you please rescind my tax cut? Will you please raise my taxes so we can get our economy moving again?" he said.

After his speech to about 1,600 people in the grand ballroom of the Waldorf Astoria, Reagan met for about 45 minutes with U.N. Secretary General Javier Perez de Cuellar and U.S. Ambassador to the United Nations Jeane J. Kirkpatrick in a hotel suite. It was the president's first meeting with the newly selected secretary general.