AS THE BRUISING struggle to be mayor of New York City heads into a second round of primaries, we find ourselves wondering why anybody would want to win it. The job is going to be less difficult over the next four years than it has been over the last two - which is to say, very nearly impossible. The city's deterioriated economic and financial condition will confront the new mayor with the same old agonizing problem: how to keep spending down and revenues up in a city with a long history of ever-increasing costs. The best that can be said for this challenge, perhaps, is that a new man on the job hasn't got much to lose, given the record of the past few years. If he nurses the city back to health, he will get all the credit. If he does no better than his predecessors, Abe Beame - and plead in his own defense, with some hope of a sympathetic hearing, that New York's problem are chronic, widely shared by other cities and to some degree insoluble.

Few public figures have been repudiated by once friendly constituents as thoroughly as Mayor Beame was on Thursday. He was unable to get even one vote out of five cast in the Democratic primary. And the issue that brought his downfall - fiscal responsibility - didn't even dominate the closing days of the campaign. For the candidates to come down to election day shouting about capital punishment reflects a level of politics that demeans a great city. New York has enough difficulties about which the mayor can do something - the financial shambles and the crime rate are the most obvious - that the city ought to have been spared a debate on an issue about which the mayor can do absolutely nothing.

The success of Edward Koch in the Democratic primary, however, does indicate the full effect of the crime issue and television advertising on politics. Mr. Koch apparently moved himself into the run off, after trailing badly, by taking a hard line against crime, spending his money on television and staying largely away from old-fashioned organizational work.Mr. Beame still had a good part of that once powerful organization behind him and couldn't even muster a decent vote in his old home district.

We have no idea of how this kind of politics will work for Mr. Koch in the runoff against Mario Cuomo, the choice of Gov. Hugh Carey. But we hope that somewhere along the line these two candidates, as well as the other two who still remain in the field - Roy Goodman for the Republicans and Barry Farber for the Conservatives - will get back to talking about New York City's real ills. For most of these ills are common to all big cities. The way they were addressed and ultimately dealt with - or not dealt with - in New York is a matter of more than passing interest to large urban concentrations everywhere.