President Carter hopscotched across half the country today, stopping here to listen to westerners discuss water policy as he sought to renew his ties with several groups of his constituents.

In his first trip as President into a region that former President Ford swept against him last November.Carter promised a roundtable discussion on water policy that the federal government would not seek to preempt state or private management of water resources in the West.

But, as was the case at a similar discussion on urban policy in Detroit on Friday, the President, for the most part, listened, offering little more than a sympathetic ear on several of the issues of most concern to westerners.

Moving steadily westward during the day, Carter passed through the farm belt, where his farm policies have drawn criticism, through the Mountain States, where his cancellation of water projects caused an uproar, and finally to California to court some of the financially powerful figures of his own party.

After meeting with western governors and speaking briefly to a meeting on the Panama Canal treaties, the President flew to Los Angeles tonight to address a $1,000-a-plate Democratic fund-raising dinner.

Awaiting Carter at the Century Plaza Hotel in Los Angeles was the largest demonstration he has encountered since he became President.

An estimated 3,000 demonstrators, waving signs, shouted and chanted as they tied up traffic in front of the hotel. The largest contingent represented Imperial Valley farmers, who are protesting their inclusion under a law limiting ownership of federally watered land to 160 acres per person.

Most of the demonstrators were goodnatured. The only incident before Carter's arrival was when a group of Jews protesting Carter's Mideast policies inadvertently became entangled with a group of demonstrators supporting the Palestine Liberation Organization. There was a spitting incident and some threats, but no violence.

Remaining a mystery throughout the day was what, if anything, lay behind the President's prediction Friday night that the United States and the Soviet Union will sign a strategic arms limitation agreement "in a few weeks."

At a Jefferson-Jackson Day fund-raising dinner in Des Moines, in a speech devoted largely to energy and agriculture. Carter startled reporters traveling with him by declaring:

"I can tell you that in a few weeks my prediction is that we will have a SALT agreement that will be the pride of our country, and following that, we will proceed toward my ultimate goal of reducing nuclear weapons in the world to zero."

Immediately after the speech, the President seemed to back off from his prediction and his press secretary, Jody Powell, quoted him as saying, "I didn't say how many weeks." Nonetheless, the seemingly offhand remark to a political dinner appeared calculated to raise hopes that progress is being made in the SALT negotiations.

Today, Carter was asked about the prediction of Sen. Bob Dole (R-Kan.) that a SALT agreement would be rejected by the Senate. "I doubt it." Carter replied before stalking off.

When the President's cross-country trip originally was planned, its primary purpose was to attend tonight's Democratic Party fund-raiser in Los Angeles. But, in the meantime, Carter's energy legislation was torn to shreds in the Senate and the White House embarked on a strategy of making enactment of an acceptable energy plan the top priority of Carter's first year in office.

Thus, at every stop on this trip, the President has sought to press his message of an impending energy crisis.

At the fund-raising dinner in Des Moines. Carter made his most direct appeal for passage of energy legislation and repeated his now familiar attack on the oil industry.

His remarks on energy, particularly his vow to fight the oil and gas companies, were enthusiastically received by an audience of about 3,000 persons at the Des Monies Veterans Auditorium. But when he spoke of a subject of even more interest in Iowa - agriculture the Democrats in the auditorium were mute as Carter asserted that progress had been made and voiced his hopes for the future.

Iowa farmers, like others in the Midwest, are not pleased with the administration's recently passed farm bill and with this year's sharp decline in farm prices. The President was greeted at the Des Monies airport by signs saying. "Nuts to $1.46 (a Bushel) Corn." And that surfaced again this morning, when Carter had breakfast with a group of farmers.

Before reporters were ushered from the room, one of the farmers - Ferris Gray, told the president of his farm program - "We don't feel it's gone for enough." Gray added that the current prices farmers receive for corn "just don't pay the bills."

From Des Moines, the President flew to Offutt Air Force Base, the headquarters of the Strategic Air Command. He was accompanied by Defense Secretary Harold Brown and White House national security affairs adviser Zbigniew Brzezinski, who flew to Des Moines Friday night so that they could be with Carter at SAC headquarters. Also present was Joint Chiefs of Staff Chairman George Brown.

The President spent more than two hours inspecting the base's command post and various types of aircraft before flying to Denver.