The quest for the Chicago area's large Democratic vote in next Tuesday's Illinois presidential primary began in earnest today in the Polish wards of the city's North Side and its affluent northern suburbs.

While Sen. Edward M. Kennedy (D-Mass.) munched on Polish pastries for breakfast and had lunch at a Mexican restaurant in his search for votes in the city's ethnic wards, Vice President Mondale, in his familiar role as chief surrogate for President Carter, courted Jewish voters in suburban Skokie.

Both men, however, were put on the defensive during parts of the day -- Kennedy because of the continuing questions about his personal life, and Mondale because of the administration's blunder in its handling of the United Nations resolution that censored Israel for establishing settlements in Israeli-occupied Arab territories.

Two other Carter surrogates -- Commerce Secretary Philip Klutznick and campaign chairman Robert S. Strauss -- also ran into a barrage of criticism today over the U.N. resolution in an appearance before the United Jewish Fund political action committee.

During an acrimonious 90-minute session, the two Carter supporters heard the administration repeatedly accused of adopting a policy of "appeasement" toward Israel's enemies. Klutznick, a Chicagoan and founder of the organization, was clearly angry and frustrated by the end of the meeting, scolding the audience for failing to find anything to praise in the president's record.

"Do your homework a little better," he snapped.

Mondale's mission today clearly was to try to pacify the angry Jewish reaction over the U.S. vote to support the resolution, which Carter later disavowed and blamed on a "communications failure" in the State Department.

In a speech to several hundred middle-aged and elderly voters in the heavily Jewish Skokie, Mondale tried to turn the followup on the issue to the president's advantage, calling it an example of Carter's honesty and courage.

He offered a lengthy, detailed explanation of how the episode came about, contrasted Carter's handling of it with then-President Nixon's behavior during the Watergate investigation and declared:

"The best way to handle a mistake is to say it's a mistake and have a leader with the courage to do it, and I'm proud that he just said so."

Mondale was warmly received. Later, however, several in the audience said they expect a heavy Democratic crossover vote in their area Tuesday for Republican presidential hoepful Rep. John B. Anderson of Illinois.

Aides in both the Kennedy and Carter campaigns here say they are uncertain how many normally Democratic voters Anderson will attract, or which of the two leading Democratic contenders would be most hurt by a large crossover vote.

Kennedy, meanwhile, got off to a shaky start this morning when he was hit with a series of questions about his personal life during the taping of an interview at a local television station.

While Kennedy tried to turn the conversation to the issues of the economy and health care, his questioner, Sandy Freeman, returned repeatedly to the themes of his marriage, the acknowledged alcoholism of his wife, Joan, and his "honesty and morality."

"I'm quite prepared to place both my honesty and my commitment to this country against any other candidate for the presidency," Kennedy said tersely.

Later, with snow pelting his campaign entourage, the Massachusetts senator visited a restaurant where he was serenaded with a Polish folk song, toured a neighborhood redevelopment project where he held a listless conversation with some of the workers and met with Hispanic community leaders at a local savings and loan.

After the meeting, Kennedy declared his support for the inclusion of projects to restore sidewalks in older city neighborhoods in federal urban aid programs.

It was not until mid-afternoon that Kennedy had a chance to talk about the larger economic issues that he hopes will give a push to his uphill campaign against Carter.

In a speech to the National Association of Neighborhoods, Kennedy denounced the president's economic policies, charging that the up to $20 billion in budget cuts the administration is expected to propose will do little to ease inflation but will cost 1.5 million workers their jobs.

The issue in the presidential campaign, Kennedy said, is whether inflation is to be fought by people "all across this nation, or whether these battles are going to be fought on the backs of the elderly people, the poor people, the women in society, in minorities.