Protests against South Africa gathered force in the 16th day yesterday, capturing the attention of key Republican senators while a son and daughter of the late Robert F. Kennedy were arrested outside the embassy in Washington.

On Capitol Hill, Sen. Richard G. Lugar (R-Ind.), the incoming chairman of the Senate Foreign Relations Committee and a staunch administration supporter, met with the Rev. Jesse L. Jackson and later told reporters that the administration "needs to do more" than it is currently doing to "manifest our strong feelings about South Africa."

Lugar and Sen. Nancy L. Kassebaum (R-Kan.) later met with South African Ambassador Bernardus Fourie to express concern about the situation in South Africa, particularly the incarceration of 21 labor leaders.

The arrests of nonviolent protesters continued in Boston and New York and spread to Chicago, where outside South Africa's consulate, Jackson's wife Jackie, an Illinois state senator and U.S. Rep. Gus Savage (D-Ill.) were arrested.

And on Massachusetts Avenue NW, Sen. Gary Hart (D-Colo.) joined the demonstration outside the South African embassy. For the first time the protest had the look and feel of youth, with jeans-clad college students spilling across the pavements and chanting in loud, impassioned voices: "Free South Africa, free Namibia, free the slums, free the ghettos. All people must be free."

Robert Kennedy's youngest son, 17-year-old Douglas Harriman Kennedy, invoked his father's words and memory during a short press conference held a few feet from the crowd of about 400 demonstrators.

"I feel my presence here and the presence of so many other students symbolizes the fact that my father's ideals for the world are still alive," Kennedy read from lined notebook paper.

Then the young Kennedy quoted his father's words on a trip to South Africa several years ago: "Each time a man stands up for an ideal or acts to improve the lot of others or strikes out against injustice, he sends forth a tiny ripple of hope . . . a current that can sweep down the mightiest walls of oppression and injustice."

Minutes later, Douglas Kennedy, his 15-year-old sister Rory Elizabeth, and a friend, Derrick Evans, 17, a student at Boston College, crossed police lines 500 feet from the embassy, walked to the embassy doorstep, linked arms and sang "We Shall Overcome." Police arrested them, and they were later charged as juveniles with the misdemeanor of congregating within 500 feet of an embassy with intent to demonstrate.

There was evidence that the antiapartheid movement's impact was spreading. As of yesterday, the total arrests in Washington stood at 28. In New York, five new arrests brought the total to 22; in Boston, six more arrests brought the total to 10, and Chicago had its first three.

Congressional aides were attempting yesterday to set up a meeting between House Speaker Thomas P. (Tip) O'Neill Jr. and South African Bishop Desmond M. Tutu, who is to meet today with President Reagan to discuss the administration's South Africa policy. Congressional aides said they were unsure whether the meeting, which O'Neill requested, could be scheduled before Tutu has to leave the country for Oslo to receive the Nobel Peace Prize.

Across the Capitol in the Senate, influential members were reacting, although foreign policy experts there predicted the result may be heightened rhetorical pressure on South Africa rather than legislation to make substantive changes in U.S. policy.

President Reagan's policy of "constructive engagement" involves trying to influence South Africa through friendly persuasion to change its apartheid system of strict racial segregation rather than using punitive measures, such as economic sanctions.

In a strongly worded letter to Reagan, Lugar and Kassebaum, who chairs the Foreign Relations Committee's African Affairs subcommittee, denounced the State Department's failure to speak out against apartheid, according to a Knight-Ridder Newspapers report.

"The unwillingness of the State Department to attack the evils of apartheid and the violations of human rights in a straightforward, understandable manner has undermined the effectiveness of whatever private efforts we are making," the letter said. "We believe that your . . . repugnance of the system of apartheid has not been as forcefully applied to the South African situation as it should be."

Kassebaum refused to discuss the "private communication" with Reagan and said in an interview that she is not in disagreement with the "basic concept" of the Reagan policy. "But things must be done to show we will not accept everything that is done by South Afica without saying 'Wait a minute, this policy is repugnant.' "

Some members of Congress have proposed economic sanctions against South Africa until apartheid is ended, and on Wednesday 35 conservative Republicans told the South African ambassador they would support sanctions unless that country demonstrates rapid progress toward ending its system of racial segregation. But others were skeptical.

"If it would really end apartheid, I would be for them sanctions , but I doubt that it would," said Sen. Larry Pressler (R-S.D.), a member of the Foreign Relations Committee. He expressed support for the demonstrations and said he agreed generally with the tough statements of the House Republicans, but that if the United States applied sanctions to all countries with which it disagreed, "we wouldn't trade much."

"We'd like to pass some real zinging legislation, but there is so little you can do," said an aide to Sen. William V. Roth (R-Del.).

"I don't think economic sanctions would ever take place in this country," whether Democrats or Republicans are in power, said one Senate staffer. "It's a boomerang," because the United States depends heavily on South Africa for strategic metals, the staff member said.