President Reagan's proposal to sell sophisticated military equipment to Saudi Arabia has dealt the Democratic Party an issue it can neither refuse to exploit nor afford to overplay.

The arms package, which Israel opposes for security reasons, gives the Democrats their first clear opportunity of the Reagan presidency to woo back the nation's Jews, who abandoned the Democratic Party in droves last year.

Democratic National Committee Chairman Charles T. Manatt, eager to capitalize on the controversy, has been meeting privately with Jewish leaders and lobbying extensively on the Hill.

But he has taken a low-key, behind-the-scenes approach to the issue out of fear of alienating Jewish leaders, who do not want domestic political calculations to endanger the bipartisan nature of congressional opposition to the proposed sale.

"It's something that has to be handled gingerly," says Manatt, a Californian who has been DNC chairman for the past seven months. He and other DNC staffers say they have been cautioned by friends in Jewish organizations to refrain from politicizing the issue too heavily.

In the Senate, where the showdown on the proposed sale of radar (AWACS) planes and other state-of-the-art military equipment is expected next month, 18 of the 50 co-signers of a resolution opposing the sale are Republican. "Obviously, we don't want those Republicans to feel like they're sponsoring a Democratic resolution," said one spokesman for a national Jewish organization who asked not to be identified.

The Democrats are satisfied that they have walked the right line on the issue. "We're on record opposed to the arms package, but we aren't going around to Jewish groups saying, 'Ha, ha, the Democrats are on the right side of this one,' " said Lynn Cutler, a DNC vice-chairwoman.

Manatt denounced the sale this spring, and the executive committee of the DNC followed in the summer with a resolution of opposition. Since then, Manatt says he has been personally in touch with more than two dozen congressmen and senators on the issue.

He has also been instrumental in putting together an umbrella organization, the Coalition for Strategic Stability in the Middle East, that will coordinate the final thrust of lobbying efforts to block the sale. But when the group holds its inaugural press conference Monday, Manatt will be in the background. The group's bipartisan co-chairs will be J. C. Turner, president of the International Union of Operating Engineers, and Clifford Case, a former Republican senator from New Jersey.

"We have deliberately not leaped out in front and made this a big DNC issue," said DNC staffer Bernard Aaronson. "We don't want to jeopardize the defeat of the sale. If we were only interested in reaping some political hay, I think you might have seen a DNC ad campaign and other efforts to identify our name with the opposition."

Aaronson's comments reflect a problem that Manatt faces as the Democrats adjust to the unaccustomed role of loyal opposition. The party must oppose administration policy and offer alternatives, but that effort in and of itself can sometimes prove counterproductive.

"Manatt should speak what he believes, but I don't think he ought to make it the AWACS issue a matter of Democratic policy," said Joseph Rauh, longtime liberal leader. "If he did, you could have Reagan lining up Republican votes by saying, 'Well, look at what the other fellows are doing.' "

Democrats learned that lesson the hard way this year. They spent six months vainly opposing the president's economic program. On the few occasions they appeared to have victory within grasp, Reagan was able to counterpunch effectively by portraying the vote in question as an acid test of his administration's credibility and mandate.

Still, the AWACS flap gives the Democrats their best chance since Reagan assumed the presidency to rebuild bridges to a highly valued constituent bloc. In 1976, an estimated 75 percent of the nation's Jews supported Jimmy Carter for president; in 1980, that figure fell to 45 percent.

Jewish fund-raising, long a mainstay for the Democratic Party, showed a commensurate decline over the four year span, Manatt said.

Jewish leaders who supported Reagan last year are not unmindful of the domestic political risks the president has taken on. Some 38 leaders wrote him a letter this month urging him to reconsider his position. At the same time, however, they reiterated their support for his administration.