Republicans and Democrats agreed yesterday that double-digit unemployment would hurt the GOP at the polls on Nov. 2, but they differed on whether it would be just an added burden or the straw that breaks the elephant's back.

As Democrats from coast-to-coast proclaimed September's 10.1 percent unemployment rate announced yesterday as the final proof of the failure of Reaganomics, President Reagan and his spokesmen launched a vigorous counteroffensive.

Reagan laid the blame on decades of excessive spending and taxing by Democratic congresses, but Democratic National Chairman Charles T. Manatt said it was a signal to "change course" from Reaganomics.

Democratic campaign planners expressed confidence that Manatt would be proved right when he said that crossing this "new threshold of disaster" would finally exhaust "the patience of the American people" with administration promises of imminent economic recovery.

But Roger Stone, a campaign consultant to Republican candidates in more than a dozen states, said the stock market boom and the decline in inflation and interest rates mean "it's not the gloom-and-doom picture that Democrats would like to paint."

Patrick Caddell, a leading Democratic pollster, said that, coming in the early October period when most voting decisions are made, the announcement would likely "break the psychological barrier" and push wavering voters toward the Democrats.

But Republican media consultant John Deardourff called it "a 48-hour bad story" that would be "no way fatal" to most GOP candidates.

Whoever proves right in assessing the implications, there was no question that Democrats were seizing the moment to increase the volume of their verbal assaults on the administration and its supporters.

Caddell said many of his clients had prepared new commercials in advance, anticipating the bad news, and put them on the air last night "to piggyback on the free media" focus on the jobless figure.

The Association of House Democratic Press Assistants sent out a memo last week suggesting that incumbents visit unemployment offices yesterday to provide the press with shots of them "talking with the unemployed, soliciting first-hand the stories of the hardships they're facing."

Some candidates were more imaginative than that. Lane Evans and William Gluba, Democratic challengers in Illinois and Iowa congressional districts on opposite banks of the Mississippi River, met in midstream aboard a ship temporarily christened "Solidarity" to pledge their support for a public-service jobs program.

Adam Levin, a Democratic congressional candidate in New Jersey, sent a tongue-in-cheek message of "congratulations to the Reagan administration. You now have a perfect 10. Unfortunately, it's the unemployment rate."

There was little jesting among Republicans on the issue. Rep. Hal Daub (R-Neb.), a conservative freshman facing an unexpectedly stiff battle for reelection, said he would not "attempt to lessen the serious nature of the problem."

Daub, anticipating the unemployment report, made public a letter he sent the president last week in which he urged Reagan "to submit to Congress a budget that treats our economic problems as if they were the most significant threat this country faces, because they are."

But, like other Republicans, Daub tried to turn the blame on the Democrats, saying, "We are paying for their reckless spending."

Manatt and Rep. Morris K. Udall (D-Ariz.) sought to dispel the charge that Democrats have no alternative during a press conference at the headquarters of the American Federation of State, County and Municipal Employes that was beamed to television stations around the country.

They endorsed emergency housing and public works programs they said would create a half-million jobs, and Udall also called for repeal of the pending 1983 income tax cut of 10 percent.

Those measures were plugged by many Democratic congressional candidates, but aspirants for state office also played off the news to plug job programs of their own. In California, Los Angeles Mayor Tom Bradley (D) scheduled a radio address for today to announce a plan his press secretary said would "create hundreds of thousands of new jobs."

California was one of four major states where unemployment declined slightly in September. In Florida, another of those states, Gov. Bob Graham (D) said the reduction from 7.7 to 7.5 percent was a tribute to his efforts to diversify the economy of the state.

Meanwhile, in Texas, where joblessness jumped from 6.7 to 8.4 percent, Gov. Bill Clements (R) chose to focus on the fact that "good economic news -- declining interest rates and a soaring stock market -- greeted us this morning."

His opponent, state Attorney General Mark White (D), said that while Clements may think "the jobless situation is not significant, I say the fact that more than 600,000 people are now out of work in this state is unconscionable."

Looming over all the partisan rhetoric was the unanswered question of whether the patience and hopefulness pollsters and politicians have found all year among the majority of voters will be snapped by the latest unemployment report.

Robert Teeter, a leading Republican pollster, said the strange mixture of economic news, with a stock market boom coinciding with the highest unemployment since 1940, would serve mainly to sharpen the economic debate for the final three weeks of the campaign.

Both Teeter and Deardourff said they thought any advantage the Democrats gained yesterday would be short-lived, and both focused on the economic policy speech Reagan is planning for next week as the critical test of Republicans' ability to contain the damage.

"The whole question," Deardourff said, "is whether the president can convince people that everything else--inflation, interest rates, taxes, spending, regulations -- is moving the right way, and unemployment is just the last stubborn holdout, which must follow."

"The Democrats," Teeter said, "are saying that things are worse, so vote Democratic. But if Reagan can renew the faith that things will get better, their attack may not work."

Stone, the third GOP consultant, said that in most of the states where he is working, "there was a dip in support for Reagan and the Republicans two weeks ago, but it was a slight dip and most of our candidates are still in pretty good shape . . . ."

But all three of the Republicans agreed that their clients are at risk either if Reagan is seen as being "unaware of the suffering of the unemployed," as Deardourff put it, or if voters who basically agree with Reagan's goals and strategy still decide to express their impatience by voting Democratic this fall.

Stone said this kind of "send him a message" vote was a greater threat to Reagan than outright repudiation of his conservative program.

Caddell said the Republicans were whistling in the dark.

"Every campaign waits for an event that becomes its galvanizing moment," he said, "and this is it for 1982. Both the political community and the media have been waiting for this moment, and what voters are hearing this weekend will confirm their inclination to resolve their doubts in favor of the Democrats. The timing could not have been worse for the Republicans. This is the point of the campaign where most people decide. They would have been better off if the news had come in August or September."