Thousands of Americans marched through the frigid streets of Washington yesterday and hundreds more congregated in the warmth of churches, libraries and auditoriums for celebrations to mark the 53rd birthday of slain civil rights leader Martin Luther King Jr.

Throughout the city--from the slush-filled streets of old Anacostia to the expansive and starkly modern interior of the Hubert H. Humphrey Building, politicians, civil rights leaders and entertainers used the occasion to rail against President Reagan's policies, which many said were turning back the clock on King's dream.

"Can you imagine: We find ourselves fighting against an administration that is trying to give tax exemptions to schools that segregate," said D.C. City Council Chairman Arrington Dixon at a ceremony at the Martin Luther King Library downtown. "Martin Luther King is probably turning over in his grave."

The major event of the day was the march on the Mall, in which thousands took part. Park Police estimated the crowd at 10,000 to 15,000. The march's organizers said 50,000 participated. The major attraction there was Stevie Wonder, the Motown singer and black folk hero who for two years has crusaded to make King's birthday a national holiday.

"I know you've been standing in the cold for a long time, but I hope your spirits are warm," Wonder told the shivering crowd. "I hope your spirits are hotter than July." Wonder's ode to King, "Happy Birthday," is one of the songs on an album released two years ago, "Hotter than July."

He compared the audience to members of an orchestra, saying, "Dr. King left an unfinished symphony which we must finish. We must harmonize our notes and chords and create love and life . . . . We need a day to celebrate our work on the unfinished symphony, a day for a dress rehearsal for our solidarity."

Sponsors of the march said more people were expected but the inclement weather apparently took its toll. Last year, Park Police estimated the crowd at 35,000, and 60,000 had been expected this year. Donna Brazille, who had organized people in southern states for the trip to Washington, said yesterday that buses carrying about l5,000 were canceled because of the weather.

Unusually cold and snowy weather also hampered observances elsewhere. In Atlanta, King's home town, snowstorms blocked singer Harry Belafonte and former vice president Walter F. Mondale from attending ceremonies where Belafonte was scheduled to receive the Martin Luther King Jr. Non-Violent Peace Prize.

Despite bad weather, however, there was a gospel concert in Chicago, a candlelight service in Dayton and a human rights award banquet in St. Louis.

In Philadelphia, only a third of the invited guests showed up for Mayor William J. Green's prayer breakfast honoring King because of a boycott organized by an association of black policemen, protesting what they said was discrimination and racial imbalance in the city's police department.

The impetus for much of yesterday's activity was the drive to makeKing's birthday a national holiday.

Nearly 14 years after King's death, only l0 states, including Maryland, and the District of Columbia celebrate King's birth as a legal holiday. Legislation to make the day a holiday was introduced in Congress four days after his assassination in Memphis on April 4, 1968. That proposal, sponsored by Rep. John Conyers (D-Mich.), has been renewed every session since, but has never passed in the House.

The Virginia General Assembly passed a King holiday bill in 1980, but the measure was vetoed by Gov. John N. Dalton.

Montgomery County Executive Charles W. Gilchrist yesterday proposed to the County Council that the jurisdiction observe King's birthday as a legal holiday for county government workers, who do not now benefit from the state joliday, starting next Jan. 15. A county spokesman said five members of the seven-member council asked about the proposal have said they will support it.

President Reagan said yesterday that Americans should rededicate themselves to making King's "dream come true for all Americans."

"He dreamed of an America in which our children 'will not be judged by the color of their skin but by the content of their character,' and he reminded us that 'injustice anywhere is a threat to justice everywhere.' His time among us was cut tragically short, but his message of tolerance, non-violence and brotherhood lives on," the president said in a prepared statement.

White House spokesman Larry Speakes told reporters he did not know if Reagan has taken a position on declaring King's birthday a national holiday.

Reagan's policies came under fire yesterday during a memorial at the Humphrey Building where Health and Human Services Secretary Richard S. Schweiker and Education Secretary Terrel H. Bell presided. About 400 persons attended.

There, the Rev. Leon Sullivan of Philadelphia, director of the Opportunities Industrialization Centers of America, drew a standing ovation from the audience when he declared that blacks "will not tolerate federal attempts to provide tax benefits to schools that practice segregation.

"We were in that struggle before and we will not go back to it . . . . If necessary, we will call a national strike and immobilize the nation."

Schweiker, a former senator from Pennsylvania, joined in the applause. Bell did not.

The day began with a 9 a.m. march in Anacostia, where residents east of the river staged their fourth annual Martin Luther King Jr. Day Parade. The temperature hovered around 22 degrees as bands and drill teams assembled at Good Hope Road SE for the two-mile procession up Martin Luther King Avenue.

Several City Council members rode in the parade and were joined at a rally by Mayor Marion Barry, who called for enactment of legislation to make the day a federal holiday, and criticized the Reagan administration's efforts to balance the federal budget "on the backs of the poor and blacks."

At the Martin Luther King Library, the Rev. Robert L. Pruitt, pastor of the Metropolitan A.M.E. Church, recalled how last year at this time the country was five days away from Ronald Reagan's inauguration.

Now, Pruitt said, "The poor are the ones suffering from this attempt to revive and restructure the economy . . . . Today many of you have already received your pink slips . . . Many of you started in upper-level positions only five or l0 years ago. Now you are facing downgrading or unemployment."

On the Mall, marchers were led to the Capitol steps by the Roosevelt High School Marching Band. Some of the onlookers carried picnic lunches and perched in trees along the parade route.

"This is the least that I can do for a worthy cause," said Helen Turner, a receptionist with a law firm who took the day off even though it was not a holiday at her office. "King stood for a lot; at least we can remember that."

The rally on the Capitol steps was led by political lyricist Gil Scott-Heron. He was accompanied on stage by Diana Ross, Gladys Knight and Peaches and Herb. Jesse Jackson also spoke, calling on the crowd to "launch an attack on the private economy.

"If corporate America has its roots in the suburbs, it still has branches hanging over black America," Jackson said. "We should cut off those branches so that some of those apples will be available for us."

The crowd soon became impatient, and noticeably cold. Some of the snow on the Capitol grounds had begun to melt and many of the participants got their feet wet and began to disperse.

Thousands still remained as Wonder took to the podium to conclude the rally, beginning with a moment of silence for the victims in Wednesday's crash of an Air Florida jetliner into the Potomac River.

"Many times in life things happen and we question God as to why," Wonder said. "These are not easy times, yet they are not hopeless times. We must refresh our souls and uplift our spirits and harmonize with our brothers and sisters."