Senate Majority Leader Robert C. Byrd (D-W.Va.) criticized President Carter yesterday for failing to see the American public and Congress on the need for his imperiled $10 billion fee on imported oil. But he said he believed that if the president uses his veto to try to save the fee, the Senate will sustain him.

Byrd told reporters at his regular Saturday news conference that revenue from the fee is needed as a "cushion" to ensure a balanced budget and allow a hoped-for tax cut this year.

The proposed fee is in trouble not only in Congress, which has indicated it will reject the estimated 10-cent increase in the price of a gallon of cas, but also in the federal courts, where a judge has ruled Carter exceeded his constitutional authority in imposing it.

"The president has failed to adequately convey the necessity of this fee," Byrd said. "He's undersold it, both to the American people and the Congress."

Though it seems likely now that both the House and Senate will vote to block the fee, Byrd said he felt there are enough votes in the Senate to support an expected Carter veto o fthat move.

"It's winnable," he said. "I would hope the Senate would demonstrate the courage to sustain the veto" an impose the fee.

The majority leader spend most of the hour with reporters stressing the importance of motivating the public to accept painful measures aimed at energy independence, something the Carter administration has been preaching as the moral equivalent of war for the past three years.

Instability of regimes in the Persian Gulf makes a cutoff of foreign oil increasingly likely, he said. "We must not wait for the emergency to hit. It would be economic and strategic suicide."

Failure to take even the "very modest step" of imposing the import fee would be viewed abroad as a sign of continued domestic indifference to America's energy problems, Byrd said.

But he seemed even more concerned about the need for the fee as fiscal insurance. "If we're going to have a tax cut and a balanced budget we're going to have to have a cushion," he said.

Byrd said he envisioned $6 billion to $7 billion in tax relief through a rollback in the scheduled increase in Social Security taxes next Jan. 1, with $3 billion to $4 billion in an additional general "productivity" tax cut.

Even this could be threatend by increased government spending on such items as soaring unemployment compensation if the current recession is deep, he said.

Byrd even cited promotion of the import fee to support his continued belief that Carter should debate his primary election challenger, Edward M. Kennedy (D-Mass.). The importance of the fee could be drawn out in a debate, he said. Such a head-on discussion of the issues also would enhance the chances of the Democratic Party in the fall, he added.