The Senate, in a rare display of bipartisanship on the issue of campaign finance reform, last night approved a Republican proposal to ban members of Congress from sending their constituents unsolicited, taxpayer-financed mass mailings during the year in which they are up for reelection.

The proposal, approved by a 98 to 1 vote, would also end the practice under which senators who are not up for reelection give their unused "franking," or free mailing, privileges to senators who face tight reelection contests. In an action aimed largely at the House, it would cut off mass mailings when appropriations run out and require that House members join senators in making quarterly public accountings of their mailing costs.

But these provisions appeared certain to be rejected by the House, whose members serve two-year terms and thus would be banned from using free mass mailings every other year.

The proposal was offered by Sen. Don Nickles (R-Okla.), chairman of the National Republican Senatorial Committee, as the Senate moved slowly toward final action on the Democratic-sponsored bill to overhaul congressional campaign finance laws.

Working late into the night, the Senate voted 60 to 38 to reject a proposal for broader taxpayer financing of campaigns than it approved earlier in the day, when it agreed to limited subsidies for broadcast campaign ads and other incentives to encourage compliance with voluntary spending limits.

Sen. Christopher J. Dodd (D-Conn.) also served notice that he plans to use the bill as a vehicle to force a vote on banning senators from accepting honoraria for outside speeches, starting next year. The House voted last year to ban honoraria in exchange for a large pay raise in 1991, but the Senate refused to do the same. Dodd's proposal does not include a pay raise, although others may seek to make the honoraria ban conditional on increased pay.

In the debate over mass mailings, Nickles said they are part of an "enormous advantage" that incumbents enjoy, largely at taxpayers' expense, when they seek reelection. "The frank has been abused and needs to be reformed," Nickles said.

It is no coincidence that mailing costs rise dramatically in election years, Nickles observed. In the non-election year of 1987, the combined total of House and Senate mailing costs was $63 million, he noted. In 1988, an election year, the total cost was $113 million.

While not opposing Nickles's proposal, Democrats contended that most of the reforms he sought have already been instituted in the Senate. Senate Rules Committee Chairman Wendell H. Ford (D-Ky.) said he preferred current Senate rules, under which both senators from a state are prohibited from making mass mailings 60 days before a primary or general election.

In other action on the bill, the Senate voted largely along party lines to reject a move by Republican conservatives to impose major new restrictions on political activities by labor unions, which generally support Democratic candidates.

Splitting largely along party lines, it voted 58 to 41 to kill a proposal by Sen. Mitch McConnell (R-Ky.) to force labor unions to choose between engaging in political activities and continuing to benefit from tax-exempt status. McConnell's proposal would extend to unions and other tax-exempt organizations the same prohibition on political action that currently applies to charities and other groups that also qualify to receive tax-exempt contributions.

Democrats vigorously opposed the proposal, contending it would also apply to farmer, business and recreation groups. "Would it now be illegal for the garden club . . . to commend the president for his efforts to plant trees around the country?" asked Sen. David L. Boren (D-Okla.).

Voting 59 to 41, the Senate also scuttled a proposal from Sen. Orrin G. Hatch (R-Utah) that would have required unions to notify both union members and nonmembers who contribute fees for collective bargaining that they can demand return of all money that is used for political purposes if they object to the expenditure.

Instead, the Senate approved, 57 to 43, a Democratic alternative that would codify Supreme Court rulings that bar unions from using for political purposes any fees collected from non-members to defray collective bargaining costs and require that individuals be notified of their rights to rebates.

Without regulation, unions are channeling at least $300 million into political campaigns, nearly all of it for Democrats, Hatch contended. Such unregulated or "soft" money that flows indirectly into campaigns is justly called "sewer money" and should be brought under control, he said. "I don't care if this is corporate sewer money or union sewer money; this is not good for the American political system," he added.

Boren, the floor manager for the Democratic bill, said Hatch's proposal would inflict unduly onerous restrictions and reporting requirements on unions. He contended that Hatch and others would be "rightly critical" if such requirements were imposed on small businesses.