These are not the best of times for federal District Court judges trying to get the ear of their superiors on the Supreme Court.

First there was U.S. District Court Judge Harry Claiborne of Nevada, who faces a two-year sentence on two counts of income tax violations.

Claiborne, 67, the first federal judge convicted of a criminal offense committed while on the bench, wanted to argue that a sitting judge cannot be indicted. Nice try, but no cigar, the court said in effect last week, declining to hear the case. Claiborne is appealing his conviction.

Then came Judge Alcee Hastings of Florida, who was indicted on bribery conspiracy charges. Hastings was acquitted, although a separate jury convicted his co-defendant, Washington lawyer William Borders.

Two federal judges subsequently urged that Hastings be impeached for "odious behavior," including his charges that the prosecution was motivated by racism. A judicial committee was appointed to investigate the complaints, as a 1980 law allows.

The committee wanted the evidence seen by the grand jury in its investigation of Hastings. Hastings objected, but he lost in the lower courts. Tuesday, the high court turned a deaf ear to his appeals.

Hastings could be back to the Supreme Court if disciplinary action is taken against him. MEANWHILE, IN CONNECTICUT . . .

Times are hardly better for beleaguered litigants in Connecticut. Some residents there are unhappy about delays in the state court system, where it can take eight -- yes, eight -- years for civil cases to go to trial in Hartford, New Haven, Bridgeport or Stamford. The average case takes more than five years to work its way through the system in those cities.

The group sued state officials, demanding that something be done, but state courts, including the Connecticut Supreme Court, said the judiciary could not intervene to order the legislature to finance more judgeships.

Tuesday, the nation's highest court chose not to get involved. HELP WANTED . . .

But there is some good news for Democrats who are yearning for appointment to the federal bench.

There is an unexpectedly large number of vacancies now. At last count there were 97 in all, far more than had been anticipated when Congress added 85 judgeships to the federal bench in July.

Turns out that the Reagan administration, which was given the chance to fill 40 new seats before November, got around to nominating only 30 candidates. The Senate has confirmed 11 of them -- nine for new seats and two for vacant seats.

There wasn't enough time to get all potential nominees up to Capitol Hill, one administration official said yesterday. In some cases, the background checks took longer than expected. In addition, the administration relied on GOP senators' recommendations for District Court vacancies, and those did not come in as quickly as the White House had hoped, the official said.

The next president will have at least 76 new judgeships to fill on Jan. 20, 1985, in addition to 21 current vacancies.

If President Reagan is reelected, he not only will probably get the chance to appoint a few Supreme Court justices, but certainly will have the chance to eclipse the record 265 federal judges appointed by President Jimmy Carter.

Reagan, who has filled 165 seats on the federal bench, would have an opportunity to influence the federal judiciary's philosophy and direction for decades. Given the number of vacancies and anticipated retirement rates, it is likely that by the end of a second term, Reagan will have had the chance to fill more than half the 744 federal trial and appellate judgeships.