Supreme Court nominee Douglas H. Ginsburg began his campaign for Senate confirmation with get-acquainted meetings yesterday as key senators moved to head off repetition of political recriminations sparked by its rejection of Judge Robert H. Bork.

Senators said Judge Ginsburg made a good initial impression, although most, including Minority Leader Robert J. Dole (R-Kan.), said they know far too little about him and his record to assess prospects for confirmation by the Democratic-controlled chamber.

Dole joined several senators who provided critical swing votes against Bork in urging colleagues to avoid a "rush to judgment" in any direction.

They urged that Judiciary Committee hearings begin as soon as possible but cautioned against undue haste in rushing to complete them. Arlen Specter (R-Pa.), one of the panel's swing members, suggested that Congress be prepared to postpone adjournment, set for Nov. 21 and meet through the end of the year, if necessary, to complete action on the nomination.

However, other senators said the American Bar Association has indicated that it may not complete its report on Ginsburg before Dec. 1, which could delay committee hearings. A date for the start of hearings will probably be set next week, a committee source said.

Ginsburg's relatively "blank slate," as some senators called it, appeared to be working for and against the 41-year-old conservative appeals court judge as he made his Senate visits, accompanied almost everywhere by White House chief of staff Howard H. Baker Jr.

While critics questioned Ginsburg's experience, including less than one year on the U.S. Court of Appeals for the District of Columbia, they found little to seize upon in his slim portfolio of opinions and writings, in sharp contrast to multiple targets they found in Bork's more voluminous record.

"I don't know how much opposition there will be," said Sen. Howard M. Metzenbaum (D-Ohio), a prominent early foe of Bork. "I don't know how much support there will be. I don't know much of anything at all about him."

Democrats aimed their fire at President Reagan rather than Ginsburg. Reacting to Reagan's criticism of Senate handling of the Bork nomination, Majority Leader Robert C. Byrd (D-W.Va.) accused the president of trying to "stampede" the Senate into confirming Ginsburg without proper consideration.

"No president, Democrat or Republican, is going to stampede this Senate into acting with undue haste as long as I am majority leader," he said. "I'm not delaying this nomination, but I'm not unduly hastening it."

While Byrd declined to say how he will vote on Ginsburg, he reminded the Senate that he has said he wants to support a conservative.

Sen. Dale Bumpers (D-Ark.) also accused Reagan of taking a "confrontational, combative" approach in his nomination of Ginsburg and suggested that it could be counterproductive. But he urged his colleagues not to "prejudge" Ginsburg.

This cautionary tone dominated discussions of Ginsburg on and off the Senate floor.

Soon after Specter took the floor to urge against a "rush to judgment," Sens. Howell Heflin (D-Ala.) and Dennis DeConcini (D-Ariz.), who with Specter provided key swing votes against Bork on the Judiciary Committee, made the same point.

Senators should approach the Judiciary Committee hearings "with an open mind . . . and not jump to a knee-jerk reaction," Heflin said.

Heflin, one of a half-dozen senators whom Ginsburg visited, told reporters that Ginsburg made a favorable impression. He was "more relaxed . . . less defensive" than was Bork in a similar visit last summer, he said. "This man seemed more at ease," he added.

In joining Ginsburg on his rounds, which included visits with DeConcini and Specter, Baker appeared to signal his full support, although he earlier warned Reagan that Ginsburg could face confirmation problems.

Meanwhile, White House spokesman Marlin Fitzwater said presidential aides discussed Ginsburg's Jewish faith and his relative youth before his selection.

"Certainly, the fact that he is Jewish was discussed in the preliminary meetings . . . when we were trying to winnow the original list of 30 . . . ," he said. But it was not a factor in the final choice, Fitzwater said.