A restaurateur in a two-restaurant town has discovered that by consciously catering to professional athletes and their fans he can attract enough customers to stay in business -- enough even to turn a tidy profit now and then. But what he'd really like is to take a good chunk of business from his cross-town rival. What should he do?

He could work at increasing the number of sports fans in the town. He could revamp his menu, de-emphasize sports and hope for the best. He could try to attract non-fans while still paying special attention to his sports-loving regulars. He could change nothing and hope that somehow the word would go out among the non-sports-minded that the food was pretty good.

Substitute conservative-minded whites for sports fans and the GOP for the restaurant, and you've got a working description of the predicament of the Republican Party.

The Democrats are losing their vise-grip on black voters, but the Republicans haven't figured out how to take advantage of the opportunity.

And the opportunity is real, says Eddie N. Williams, president of the Joint Center for Political Studies. The JCPS recently commissioned a survey by the Gallup Organization, and the findings were fascinating.

Black voters remain overwhelmingly Democratic (77 percent), he told a recent forum of the black Republican Council of 100. But the percentage of blacks who consider themselves "strong" Democrats is slipping: 55 percent in 1984, 49 percent in 1986 and only 41 percent in 1987.

In addition, said Williams, age is working in the Republicans' favor. Only 4 percent of black voters over 50 list themselves as either "strong" or "weak" Republicans, compared with 7 percent of those between 30 and 49. For the 18-to-29 age group, the percentage climbs to 13.

But the survey turned up something else that is guaranteed to cool GOP optimism. Only 17 percent of the blacks polled think Republicans care about their problems. Even among black Republicans, only 55 percent believe their party cares about black problems. Indeed, fully a third of white Republicans doubt that the party is interested in the problems of blacks.

Can the GOP find a way to appeal to blacks who are still in the Democratic camp, though loving it less? We are not (to resume the restaurant analogy) talking about wholesale change, just minor adjustments to the menu and a serious promotional campaign. Would even such a modest effort as that reduce the enthusiasm of whites for the party?

The GOP seems comfortable enough with those blacks who come into the party even in the absence of special efforts to attract them. But as Williams told the Council of 100, such blacks, who tend to be loners, "are often viewed as lacking influence in their own party and as being more interested in their own personal economic welfare than in the welfare of the black community."

Can blacks, many of whom might be attracted to Republican ideas and most of whom see value in a strong black presence in both major parties, be expected to come into the party if doing so smacks of self-abasement?

On the other hand, can the GOP be expected to modify its programs to emphasize black concerns when so few blacks are in the party?

One black Republican, Arthur Fletcher, who was an assistant secretary of labor in the Nixon administration, has argued that blacks ought to integrate the party the way they integrated southern restaurants in the '60s. The perception of hostility "didn't stop us from integrating the schools or hotels or lunch counters," he has said. "Why do we have to wait for an invitation . . . ?

It is worth noting that it was the influence of black southerners on white southern senators that in large measure led to the rejection of Judge Robert Bork for a seat on the Supreme Court. In other words, first the blacks entered the restaurant, then they dictated a modification of the menu.

Williams thinks the burden lies in the other direction. "With all or most of the Republican candidates publicly saying the party wants more black support," he told the Council of 100, "now is the time to put their feet to the fire by asking what the party is willing to do to attract such support."

Which comes first: the new customers or the menu change? It's something for a business-minded restaurateur -- or political leader -- to ponder.