Rep. John B. Anderson's Virginia supporters said yesterday that Dr. Milton Eisenhower will serve as Anderson's surrogate running mate in a petition drive to get his independent presidential campaign on the state's ballot.

Eisenhower, brother of the late president, has agreed to allow his name to be used until Anderson names an official running mate, the Anderson campaign said. Virginia law requires that a vice-president candidate be listed on Anderson's petitions.

Eisenhower lives in Baltimore, and is the former president of Johns Hopkins University. He could not be reached for comment yesterday.

Anderson's backers will begin today to collect the signatures of 10,003 registered Virginia voters that he will need to get on the state's November ballot.

Jeff Rothman, a campaign spokesman, confidently predicted that Anderson supporters will easily collect more than the required number of signatures -- 1/2 of 1 percent of the state's registered voters -- before the Sept. 5 filing deadline.

He and Gretchen Wylegala, Anderson's Virginia coordinator, also said they had high hopes that the Republican congressman could carry the state this fall despite its history of a conservative electorate. "It's going to be tough, but we'll win in Virginia," Wylegala said.

Their reasoning, as explained at a news conference held at Anderson's Northern Virginia headquarters in Alexandria, is that Virginia voters are known "ticket splitters" and long have championed the cause of political independents.

Citing the success of Virginia's senior Sen. Harry F. Byrd Jr., an independent, and former Lt. Gov. Henry Howell's strong showing in his 1973 independent race for governor. Anderson backers said the former GOP presidential candidate should get a good reception in the state.

"Anderson is a fiscal conservative and a social liberal," said Rothman, who compared his candidate's appeal to that of Democratic Lt. Gov. Charles S. Robb and Republican Attorney General J. Marshall Coleman.

Virginia is one of 18 states where Anderson supporters are collecting signatures on petitions to get his name on the November ballot. He has already won a spot on the ballot in 14 other states.

If Anderson candidacy really catches on -- "and we're getting calls all day long from people who want to help him," -- Rothman said recent polls show that Anderson would take more votes away from Republican Ronald Reagan than from President Carter.

But Anderson's supporters in Virginia said yesterday they don't want to think about their candidate's presidential campaign in terms of helping or hurting Carter or Reagan. "A vote for Anderson is a vote for Anderson," Rothman said.