With the books barely closed on the 1988 elections, many politicians are looking to 1989, when Virginia could elect a black governor, New Jersey must pick a successor to Gov. Thomas Kean, and New York and Chicago may have spirited contests for mayor.

Other big cities will hold mayoral elections as well. Tom Bradley of Los Angeles and Coleman Young of Detroit have already announced they will seek fifth terms, and Henry Cisneros of San Antonio, Texas, has declared he will not. Kathy Whitmire of Houston and Donald Fraser of Minneapolis are expected to run again, Andrew Young of Atlanta cannot seek re-election, and George Voinovich of Cleveland - fresh from defeat in a Senate race - hasn't said whether he will.The biggest battles could shape up in New York, where Ed Koch faces severe image problems as he bids for a fourth term, and Chicago, where acting Mayor Eugene Sawyer is likely to face a multitude of opponents if the courts clear the way for an election.

Among possible opponents, none yet declared, are U.S. Attorney Rudolph Giuliani, a Republican known for prosecuting the Mafia, tax evaders and the Marcoses; Charles J. Hynes, special state prosecutor who won convictions in the Howard Beach racial attack case; former Gov. Hugh Carey; and Manhattan Borough President David Dinkins, who is black. The nation's largest city has never had a black mayor.

Chicago politicians expect the Illinois Supreme Court to rule shortly whether a special election must be held early next year or whether Sawyer can serve until 1991, when the term of the late Harold Washington, the city's first black mayor, would have expired. Sawyer, who also is black, was appointed by an angrily divided City Council after Washington died of a heart attack a year ago, months into his second term.

Race also could be a factor in Virginia, where the Democratic nominating convention is expected to choose Lt. Gov. L. Douglas Wilder as its first black candidate for governor. Wilder has the backing of Gov. Gerald L. Baliles, who is barred by state law from seeking a second consecutive four-year term.

In New Jersey, where Kean, a popular Republican, is barred from seeking a third term, several prominent politicians from both parties are expected to run for the job. Among the likely Democrats is Rep. James Florio, who lost to Kean in 1981 in the closest election in New Jersey history.