The House Ways and Means Committee voted yesterday to cut $9.1 billion next year from some of the government's largest and most basic benefit programs.

Moving to comply with the chopped-back budget resolutions that both House and Senate have passed, the committee voted almost to elminate the $1.7 billion-a-year program of trade adjustment assistance for workers who lose their jobs to imports; to reduce dramatically the present program of extended unemployment benefits for workers who have exhausted their basic 26 weeks; to cut back Medicare, welfare and some Social Security benefits, and to defer for three months part of the cost-of-living increase in Social Security benefits that will be due next year.

The Republican-controlled Senate Finance Committee already has voted similar cuts; concurrence by Ways and Means, which is in Democratic hands, virtually assures enactment. The vote was thus an important victory for President Reagan; the benefit cuts are about a fourth of all the cuts he originally proposed in domestic spending for next fiscal year.

And as further evidence of Congress' anti-spending mood, the Senate Judiciary Committee reported to the floor a constitutional amendment to require balanced federal budgets except in certain specified circumstances in the future.

But on the new and much larger cuts that Reagan proposed last week in Social Security, Senate Democrats finally united yesterday in flat-out opposition.

Partly to shore up the Social Security system, partly to help balance future federal budgets, the president proposed across-the-board benefit cuts for all workers who go on the rolls in the future, special further cuts for those who retire before age 65 and new restrictions on disability benefits.

The Senate Democrats, after months of disarray in response to the Reagan economic program, voted unanimously in caucus to oppose the new Reagan recommendations, using "every rule in the Senate book," as Minority Leader Robert C. Byrd (D-W.Va.) put it. They may try to force a vote this week.

In a statement issued after their caucus, the Democrats vowed to "enact reforms necessary to assure the long-term solvency of the Social Security system," but did not say what specific changes they would make. They also said they would not support benefit cuts "which exceed the amount necessary to achieve a financially sound system," as some charge Reagan has.

The administration, meanwhile, backpending a little, indicated again that it would be willing to compromise on its proposals, which have stirred up the most opposition of any the president has made thus far.

All committees in Congress are now seeking, as Ways and Means did yesterday, to cut programs within their jurisdictions to conform to the fiscal 1982 spending targets set by the newly passed budget resolutions.

Not all are succeeding. The House Agriculture Committee came up about $1.7 billion short yesterday after defeating, 25 to 19, a Republican attempt to cut back the price-support programs in the pending farm bill. It will try again later. On Ways and Means, even though members gave Reagan most of the cuts he wanted, they turned down his requests to blend a long list of individual programs into two "block grants" for the states involving social services and energy aid for the poor. It also cut welfare and social services substantially less -- and other programs more -- than Reagan requested and the Finance Committee voted earlier.

Major changes approved by the committee yesterday:

The special trade adjustment benefits would be cut from about $1.7 billion to about $364 million by reducing the payments to individuals substantially, making it harder to qualify and reducing the number of weeks of payments.

Unemployment compensation would be cut $1.3 billion by repealing the so-called national trigger for payment of an extra 13 weeks of benefits after the basic 26 weeks are exhausted; recalculating how certain state triggers for receipt of the extra benefits are figured: and limiting basic unemployment benefits for people leaving the armed services to 13 weeks while making it harder to qualify.

Social Security would be cut $2.7 billion by phasing out the so-called students benefit, eliminating the $122 monthly minimum benefit for persons becoming eligible in the future and paying only half the 1982 cost-of-living increase in July, deferring the rest of Oct. 1.

Welfare for families with dependent children would be cut $760 million by sharply reducing the amount of income that a welfare client need not count in determining welfare payment levels, counting the income of step-parents when a family applies for benefits and a variety of other changes. s

These changes are less severe than voted by the Senate Finance Committee earlier.

Low-income energy assistance to help the poor heat their homes would be cut to $1.4 billion, about $825 million less than the anticipated outlays.However, the committee made virtually not cuts in the big social services child welfare and adoption programs for the poor, which the finance committee had slashed by about $800 million.