As the widow of Dr. Martin Luther King Jr. looked on, the daughter of the slain civil rights leader and two other persons were arrested yesterday during the ongoing demonstration at the South African Embassy to protest that country's apartheid policy of racial separation.

Coretta Scott King, saying she, too, is prepared to go to jail to help oppressed blacks in South Africa, watched as her daughter Yolanda King, along with Gerald W. McEntee, international president of the American Federation of State, County and Municipal Employees and the first white protester to be arrested, and Richard Hatcher, the mayor of Gary, Ind., were handcuffed and placed into waiting police vehicles, bringing to 13 the number of persons arrested in the nine-day-old demonstration.

"Not too long ago, a struggle started in Montgomery," Mrs. King said, referring to the U.S. civil rights movement. "We stand today, almost 29 years later, in solidarity with our brothers and sisters in South Africa . . . who are languishing under the worst and most inhumane conditions of human servitude."

The protest and arrests yesterday were carried out under the sympathetic eye -- and with the tactical cooperation -- of D.C. police, and in view of D.C. Mayor Marion Barry, who arrived in his limousine across the street from the embassy as the arrests were being made and declared his support for the protest methods.

"I think it's the best thing that's ever happened. I'm in great sympathy," Barry said before joining a picket line of more than 200 demonstrators who were chanting such slogans as "Reagan, Botha, you can't hide, we charge you with genocide."

Earlier in the day, Barry said there is "a big difference" between the current embassy protest tactics and similar ones used in past civil rights and antiwar demonstrations. This time, the mayor emphasized, "We're in charge."

The black organizers of the ongoing "Free South Africa Movement" have been backed by Barry and other black city leaders. A black police chief heads a force that includes black police officers who have been arresting mostly black demonstrators.

As the Massachusetts Avenue NW embassy demonstrations pick up steam -- the number of protesters grows daily, and yesterday's event attracted more than 50 photographers, TV crews and reporters -- the police and protest organizers huddle at the scene to map out the fine points of the demonstration.

On Wednesday coordinators for the Southern Africa Support Project, in charge of arranging each day's demonstration, successfully appealed to police to let three demonstrators get closer to the embassy before being arrested.

Police normally keep demonstrators 500 feet away from any embassy and had been arresting those who crossed police barricades set up about a block south of the South African complex. On Wednesday, though, police said there had been "a change" in plans.

"The coordinators want to have the people who are going to be arrested to be as close to the embassy as possible," Lt. Donald B. Thomas announced to a group of reporters.

"We will escort them up to the embassy with the uniformed Secret Service . . . and they will do something outside that will force us to make the arrests."

On Wednesday as well as yesterday, the demonstrators were arrested when they refused to leave the steps of the embassy after being informed they would not be allowed entry without an appointment.

Police officials said it is not unusual for them to be so closely involved in the staging of a demonstration so as to avoid unnecessary confrontation.

South African Ambassador Bernardus Fourie, however, has complained about the "violation" of his embassy's sanctity, particularly the Nov. 21 sit-in that launched the protests. He told reporters Wednesday that the protesters should be kept 500 feet away.

While the Southern Africa Support Project is in charge of lining up picketers each day, the actual decision on who is to get arrested and when is left to "a rather fluid group" of the TransAfrica lobbying leaders and D.C. Del. Walter E. Fauntroy, president of the Black Leadership Round Table.

"It's a question of balancing schedules and of not having all the big name people go at once," said Cecelie Counts, a TransAfrica staff member. "We're talking about a protracted campaign -- we're prepared to be out here as long as it takes."

The "Free South Africa Movement" has set four goals for its anti-apartheid protest. Securing the release of 13 South African labor union leaders -- arrested earlier this month -- is a top priority.

It also wants the minority white South African government to release several other black leaders, including two who have been imprisoned since the early 1960s, and to negotiate with blacks on their grievances and on sharing power.

Finally, demonstrators want to end what the Reagan administration calls "constructive engagement" with South Africa, a policy whereby the U.S. government remains opposed to apartheid while improving relations with the country in the hope of gaining reforms over a longer period of time.

Rounding up people willing to picket or be arrested has been pretty easy, organizers say, because groups opposed to South Africa's apartheid policies have been in place for some time. These existing coalitions are being called on to help put together similar demonstrations Monday at South Africa's 13 consulates in this country.

A police spokesman said King, McEntee and Hatcher were each charged with "congregating within 500 feet of an embassy with intent to demonstrate and failure to disperse." He said all three would spend the night in jail.