A quick kill of John Anderson was a high priority for the Carter campaign. To no avail.

Anderson has put money in the bank, established himself as a significant factor in key states and virtually cleared the hurdle of inclusion in the first presidential debate. His candidacy now presents a problem for the president -- and to serious voters.

No one can doubt that Anderson works chiefly against Carter. The congressman from Illinois has almost no standing in the western states comprising Ronald Reagan's base, nor in the oil states of the South, where Reagan threatens Carter's base.

His strength is in the industrial states of the Northeast and Middle West where Carter must do or die. Polls in those states show that about 7 out of 10 Anderson votes come out of Carter's hide.

During the summer months, Anderson floundered badly. And the opportunity to finish him off fast presented itself. The Carterites found in the presidential debates an instrument for shoving Anderson down the tubes.

They took the position that they wanted first and foremost a head-to-head debate with Reagan. Then they would allow Anderson, and all other candidates, into the debates.

The League of Women Voters, which had sponsored the debates in 1976 and therefore had a leg up for this year, tried to set neutral standards. The League was to admit to the first debate any candidate with a showing of above 15 percent among decided voters in the national polls. The Carterites immediately began shooting at the League. They spread the word that the League was biased in favor of Anderson. At one point the White House even accepted an invitation from the National Press Club to debate Reagan one-on-one. When Reagan insisted on abiding by the League rules, the Carter camp implied the president would stay out of the debates if Anderson were included.

Nobody should doubt the prime purpose of that skirmish. The calculation of the White House was that if Anderson missed the first debate, he would fade from sight, thus giving the president a clear shot at Reagan. Nor should anybody take offense at that calculation. It represented the best strategy for the president. It just didn't work.

During the past 10 days, on the contrary, Anderson has rebounded. He overcame direct pressure from the president and Gov. Hugh Carey to win the endorsement of the Executive Committee of the Liberal Party in New York. That foreshadows a large Anderson vote in that state, in Connecticut, Massachusetts, New Jersey and Pennsylvania.

The Federal Election Commission has sustained Anderson's claim to government financing for any party that wins over 5 percent in the polls. He can borrow against that chance, and there is now a strong prospect he will have several million dollars available for television advertising next month.

Finally, the poll results have been pouring in. So far, Gallup, Harris, Roper and Yankelovich have put Anderson at, or above, the 15 percent figure. Though the sampling continues, the League will almost certainly include Anderson in the first debate. But what will the president do? During the primaries he walked away from debates with Sen. Kennedy and paid no price. It must be tempting to try the same tactics against Anderson. p

But the second time around is different. The president would alienate the Anderson supporters he might yet win over. He would hand Reagan a powerful issue -- the charge that Carter favors debates when he's behind but not when he's ahead, and that he approaches even public dialogue in the cynical spirit of narrow self-interest.

The president's problem expresses the problem of many serious voters. It is hard to see how Anderson could win the elections, or even carry many debates. A vote for him tends to be a vote for Reagan. So many who like Anderson but find Reagan intolerable will be tempted to hold their noses and vote for Carter.

Wait and see is almost certainly the right prescription for those voters. On the big problems -- on energy, the economy and national security -- Anderson has advanced far more sensible programs than Carter or Reagan. With Anderson in, the debates are for the other candidates a chance to measure up to higher standards.

Reagan may erase the impression of irrational personal convictions that seem to underlie his goofs. Carter may make it possible to support him without holding the nose to the point of self-suffocation. And if they both continue to look as bad as they have been looking, not a few voters will find in Anderson a means to signify to the two main parties the price the country pays by allowing turkeys to be nominated.