Blacks Climb Steep Political Ladder. Despite Mayor Sawyer's defeat in Chicago, blacks are breaking new electoral ground. RUNNING FOR OFFICE

WHEN Chicago Mayor Eugene Sawyer was beaten in Tuesday's Democratic primary, he became the first black mayor of a major city to lose to a white politician. The rate of growth in the number of black elected officials in the United States has slowed recently. Even today, they represent only 1.5 percent of the nation's elected officials.

But in a few spots, black politicians stand to break new ground in high-visibility offices in 1989. For example:

In Virginia, Lt. Gov. Douglas Wilder has a strong shot at becoming the first black governor elected to office in this country.

In New York City, David Dinkins has a chance to become the first black mayor of the nation's largest city.

In the Democratic Party, one black political operative has already made history. Ron Brown, a one-time top aide to the Rev. Jesse Jackson, became party chairman earlier this month - the first black to head a major political party.

Wins in Virginia and New York, says Frank Brown of the Joint Center for Political Studies in Washington D.C., ``would show that blacks are able to win in centers of power and to win votes from other ethnic groups.''

Beyond these benchmark races, says Robert Holmes of the Georgia House of Representatives and a political science professor at Atlanta University, ``I see the opportunity for some marginal gains.''

Blacks are expected to hold onto important mayoral posts this year in Los Angeles, Detroit, and Atlanta. Those prospects appear increasingly dim in Chicago.

At press time, unofficial returns showed Mayor Sawyer losing by a 43 percent to 56 percent margin to Cook County prosecutor Richard M. Daley, the son of the powerful former Mayor Richard J. Daley. Mr. Daley now faces the unofficial Republican primary winner, Edward Vrdolyak, who launched a write-in candidacy a week ago, as well as another black politician, Alderman Timothy Evans, in the April 4 general election. Mr. Evans is making a third-party bid but is considered an underdog.

Mr. Sawyer lost not because he was black, but because of the peculiarities of his position, political experts say. ``He was everybody's second choice,'' says Richard Day, head of a strategic research company in Evanston, Ill.

Sawyer was hastily installed into office by an unlikely coalition of mostly white aldermen after the passing of Harold Washington, the city's first black mayor. His appointment made it difficult to appeal to blacks, and the soft-spoken acting mayor did not have Mr. Washington's charisma to keep together a disparate group of blacks and liberal whites.

Black mayors have lost to white challengers in several mid-sized cities, such as Charlotte, N.C.; Flint, Mich.; and Tallahassee, Fla., but never in the nation's largest cities until now.

In Virginia, meanwhile, Mr. Wilder has already won a statewide race to become lieutenant governor in a state only 17 percent black. He carried 44 percent of the white vote. In this year's gubernatorial race, no serious opposition to him has appeared for the Democratic primary. The general election should be closely fought, although Wilder may be helped by internal battles among Republicans during the primary, notes University of Virginia political scientist Larry Sabato.

His path to political success is strikingly different from Jesse Jackson's, for example. Where the Rev. Mr. Jackson's presidential support remained among black voters and a slice of white liberals, Wilder has won over a wide spectrum of white voters. ``I would say they represent alternative views of the future for black politicians,'' Dr. Sabato says.

New York City could find itself with a black mayor for the first time in its history. Mr. Dinkins, president of the Manhattan Borough and the city's highest-ranking black official, announced two weeks ago he would run against current Mayor Edward Koch.

Mr. Koch is calling this race his toughest electoral challenge yet. Koch's aggressive, combative style and his ability to generate news are in sharp contrast to Dinkins's low-key approach. But it isn't clear that Dinkins has the wherewithal to build a strong base in the black community, political observers say, because other issues divide blacks.

These divisions are not limited to New York City or Chicago, and are a natural political phenomenon, says Wilbur Rich, author of a new biography on Detroit Mayor Coleman Young. As black political participation matures, race becomes less important.

This phenomenon helps explain why Mayor Young faces his biggest challenge from another black, City Council President Erma Henderson, why two established black figures - former Mayor Maynard Jackson and Fulton County Commissioner Michael Lomax - are battling to succeed Atlanta Mayor Andrew Young, and why Los Angeles Mayor Tom Bradley is expected to win a fifth term this year with the backing of a large number of white voters.