THE U.S. AUTO INDUSTRY is beginning the year in bad shape. As the moguls in Detroit watch their sales figures plunge, consumers continue to be avid buyers of fuel-efficient imports, especially Japanese cars which have benefited from the drop in the price of the yen against the dollar.

Thus it should come as no surprise that the Carter administration, fearing a protectionist backlash during the election campaign, has been begging the big Japanese auto companies to follow Honda's lead and build factories in this country.

In a carefully phrased speech to the New York Economic Club last week, Carter's trade representative, Reubin Askew, went public for the first time with that kind of invitation. He welcomed Honda's announcement of a projected $200 million plant in Columbus, Ohio, and in effect said -- let's have more.

Trade officials say privately that a step-up of Japanese car production in this country could defuse a growing demand to impose stiff import quotas on Japanese cars. Askew carefully avoided any threat that quotas would be imposed if such investments are not forthcoming.

But Douglas Fraser, president of the United Auto Workers, pulls no punches. He insists that Japan has penetrated the U.S. market so deeply that it now should be required to build plants here -- as, indeed, West Germany's Volkswagen has done.

Japanese manufacturers last year exported to the United States 1.8 million cars and trucks, or $8.1 billion worth (76 percent of the import market), and an additional $1.2 billion in parts, for a total of $9.3 billion in automotive products. Because U.S. exports to Japan were a paltry $200 million, the Japanese automotive surplus was a staggering $9.1 billion.

That almost precisely equals the U.S. overall trade deficit for Japan in 1979. Their exports here totaled about $26 billion and their purchases $17 billion, for a U.S. deficit of $9 billion.

"IF THEY DON'T build plants here, I would have no hesitation to hit them with import quotas," says Fraser. His bluntness is a measure of the fear in a union that used to be a leader in advocating a free-trade philosophy. Fraser now believes that any foreign company selling more than 200,000 cars a year in the United States should be required to build some of them here. Fraser insists that would be less protectionist than strict quotas or high tariffs.

Carter administration trade officials don't buy such an arbitrary standard. But with total imports now running at about 23 percent of the market, they recognize that protectionist forces will gather strength. As domestic sales move down from 10.6 million to probably 9 million or less in a stagnating economy this year, the Japanese share of the market will continue to swell in absolute and percentage terms.

The deputy ambassador for trade affairs, Robert Hormats, explained all of this to Japanese government and industry leaders on a recent visit to Japan, and came away with the feeling that Toyota and Nissan (maker of the popular Datsuns) are looking more closely at the possibility of investing here. Specific plans by both companies for truck production already appear to be in an advanced stage.

The big problem, of course, is that American companies -- belatedly but firmly convinced that there really is an energy shortage and that people are demanding small, efficient cars -- have laid on big plans to produce them.

In a recent interview, Treasury Secretary G. William Miller said that American producers, stung by rising imports, "are going to counterattack by investing $80 billion in new facilities by 1985." Will that lead to surplus capacity?

TO ENCOURAGE the Japanese, Fraser said that he would offer them the same deal that helped VW get started at its big Westmoreland, Pa., plant, which produced 160,000 VW Rabbits in 1979. Fraser gave VW three years to catch up with Detroit wage scales.

For Toyota and Nissan, which together exported 1.1 million cars and trucks to the United States last year, it's a tough decision. They recognize the prudence of defusing protectionist sentiment here, which might get out of hand in an election year.

But several of the major Japanese companies have planned new expansions at home. And then there is the question of running into excess capacity for small cars three or four years down the road. The Japanese also worry a bit about quality control of their products, and about nonfinancial arrangements with American unions.

But the VW experience is encouraging. The Rabbit apparently has lost none of its appeal to American consumers -- and, so far as anyone knows, has retained in Pennsylvania its essential German-style quality. And the company relationship with the Fraser union has seemed peaceful enough. Adding it all up, American officials feel that the leading Japanese companies rather reluctantly will take the investment plunge into the U.S. automotive market.