President Reagan's forthcoming budget has a special problem ahead of it in the Republican Senate. All of his tax proposals and most of his big budget cuts will have to go through the Finance Committee. And fully 10 of Finance's 20 members are up for reelection this year.

Finance is thus much more exposed politically than the Senate at large, where only a third of the seats will be up for grabs in November.

The chairman of Finance, Sen. Robert J. Dole (R-Kan.), is not himself up for reelection, but the reelection problems of the other committee members are nonetheless partly his problem.

"In an election year, your hearing improves," he said only half humorously the other day. "Sometimes you hear things that aren't there. I may not even be able to get a quorum."

Finance has jurisdiction over all tax matters--and on taxes it seems likely to be in a double bind. On the one hand, no member of Congress looks forward to voting for higher taxes in an election year, and the president is asking for some tax increases. But on the other hand, these will not be enough to keep the deficit from soaring beyond the levels many members regard as prudent, and there is thus pressure to increase taxes even more.

Finance also has jurisdiction over many of the major federal benefit programs, the so-called entitlement programs the president is proposing to cut. Medicare is among these, as are Medicaid and Aid to Families with Dependent Children. Here again the committee will have to choose between taking some money away from large numbers of possible voters or letting the deficit rise.

Six of the Finance Committee's 11 Republicans are up for reelection: William V. Roth Jr. (Del.), John C. Danforth (Mo.), John H. Chafee (R.I.), John Heinz (Pa.), Malcolm Wallop (Wyo.) and David Durenberger (Minn.). Four of the nine Democrats are also up: Lloyd Bentsen (Tex.), Spark M. Matsunaga (Hawaii), Daniel Patrick Moynihan (N.Y.) and George J. Mitchell (Maine).

Some of the senators profess to look forward to the year with some relish. "We'll be in the cockpit," said Moynihan, who is under attack by the National Conservative Political Action Committee and faces a challenge from former representative Bruce Caputo (R-N.Y.).

More typical is the cautious phrasing of Bentsen, whose state is strong for Reagan and tends to be conservative generally in its politics.

"In general terms, I believe that a time of recession is a poor time to be talking about tax increases of any sort," Bentsen says.

"But if the president calls for additional tax increases, or if the Republican majority on the Finance Committee begins consideration of the tax increases the president proposed last fall, I will certainly give the proposals careful consideration."

An aide to Roth provided a translation for such remarks: "If you have 10 guys up for reelection, you are going to have a heck of a time getting anybody to do anything in public."

Within this extremely delicate framework, Dole is conducting an intense campaign to line up backing for a tax increase of some sort. He has been warning of the dangers of "triple-digit deficits in 1983 and 1984 that are just not acceptable economically or politically."

Dole's strategy for building up momentum for a tax increase this year involves three steps. The most critical is somehow to have the president be neutral on the issue; if he actively opposes it, prospects for a congressionally initiated tax increase larger than he will recommend are close to nil.

The second involves getting the budget resolution on which the full Congress will vote this spring to include a requirement that new revenues be raised; this would place the Finance Committee under the gun of the budget act. The third would be to get action as early as possible in the session in order to keep the issue as far away in time as possible from the election.

Although the prospects for Finance Committee approval of a tax increase are tight, they are not entirely out the window. Even conservative Sen. Harry F. Byrd Jr. (Ind.-Va.) is grumbling that the 1981 tax cut bill went too far, and he has indicated a willingness, for example, to tighten or kill altogether the controversial section of the bill allowing corporations to buy and sell tax breaks.

Byrd, however, does not face the prospect of a reelection fight; instead, he has decided to end his Senate career this year.