Members of the National Anti-Klan Network began a 24-hour vigil yesterday in front of the Justice Department, demanding speedier government prosecution of racially motivated crimes.

The organization, which monitors, documents and protests racial violence, chose the week of the 20th Anniversary March on Washington to highlight what it contends are federal delays in prosecuting alleged incidents of violence by Klan members.

"We're here because we want the Justice Department to prosecute over 1,100 cases we can document of Klan violence," said network director Lyn Wells, who said that yesterday the group gave Daniel Rinzel, head of the department's Criminal Civil Rights Division, a 15,000-signature petition calling for the prosecutions.

"The Reagan administration has dealt with only 29 cases," Wells said. She contended that the administration has been leaving such cases up to state governments to prosecute, and that the states have not been active in pursuing indictments.

A Justice Department spokesman, Mark Sheehan, said yesterday that "it's absurd to say we don't prosecute cases of racial violence. We have no policy of deferring to the states in cases of racial violence. Our firm policy is to prosecute every case of racial violence where we have jurisdiction and enough evidence to convict."

At the microphones set up in front of the Justice Department building, a succession of speakers offered accounts of Klan activity.

Warren Cokley, a black man who is married to a white woman, said five white men broke into his Tallapoosa, Ga., house last February and assaulted him, leaving him with skull and brain injuries. Mary Joyce Carlson, a white woman, said that in 1978, a rifle-carrying posse of Klansmen fired on the car in which she and a black woman were driving home after a civil rights demonstration in Tupelo, Miss. Julia Chaney Moss talked about her brother, James Chaney, who was killed along with two other white freedom riders--Andrew Goodman and Michael Schwerner--in l964.

Anne Braden, who said she was was charged with sedition by the state of Kentucky in 1954 for selling a house in a white neighborhood to a black couple, said much protest is still necessary to stop violence against ethnic groups in America. "But if people are made aware, organize and make things move, they can influence and change public policy. That was the great lesson we learned in the '60s, and that's why we're here again 20 years later."

Southern civil rights activists formed the Anti-Klan Network in 1979 at the suggestion of the Southern Christian Leadership Conference. Representatives of SCLC, the National Organization of Women and the National Council of Churches sit on the network's executive committee.