ITALY'S INFINITELY cautious Christian Democrats, who have dominated every government for 30 years, now face a double dilemma. First they have to find some new accomodation with the Communists, who want seats in the Cabinet. But beyond that, these Christian Democrats - an ingrown circle of career politicians - have to find a way to govern an increasingly tense and divided country. The United States is now leaning on them publicly not to bring the Communists into formal participation in the government. But political strength is now so narrowly divided, in the country and in the Parliament, that no government at all can be formed without some substantial degree of Communist support.

While the slow dance goes on among the party leaders, out in the streets the fighting among political extremists is on the rise again. There has been an ugly series of attacks on public figures. Italy's inflation rate and unemployment rate are both the highest of any major developed country's; industrial production and investment are dropping rapidly. The Communists are calling for a government of national unity. But it ie open question whether that kind of cabinet, embracing all major parties, would make decisive government easier.

The American objections usually begin with references to the effects on NATO, but that's only a part of it. The French elections will be in March, and the parties of the left are showing great strength in the opinion polls. The quarelling among them has made it highly unlikely that the Communists will be in the next French government, but success for the party in Rome would confer - in at least some French circles - a certain reflected respectability and acceptability on the party in Paris. There is also Italy's economic program, designed mainly to bring inflation under control. The International Monetary Fund has been giving Italy financial support, but only on condition that it maintain a comparatively restrictive economic policy to try to hold prices down. That, of course, will become much more difficult if the Communists come into the government carrying all of the pent-up expectations of their followers. Governing Italy currently means presiding over a drop in the standard of living, not a comfortable process regardless of who is in the Cabinet. but the uncertainty over the economy and the uncertainty over future Communist influence tend to aggravate and compound each other.

It is quite wrong to speak of a possibility that the Communists will come to power in Italy. They have already come to a very substantial degree of power, especially since their gains in the 1976 election; they share widely in the actual governance of Italy. They brought down the Andreotti government by a gesture of their support. The immediate prospect now is for a long and intricate parley in which the Christian Democrats will give a little ground while still seeking to avoid that final, formal recognition symbolized by Cabinet appointments. Whether they succeeded is entirely unpredictable.

Some Italians have vehemently protested the State Department's statement last week declaring American disapproval of Communists in European governments. They called it interference in their internal affairs. But that's a bit disingenuous. If the Carter administration had said nothing, its silence would have been cited in the same quarters as evidence of acquiescence. Yet, having made that declaration, what else - if anything - ought the United States do? As a practical matter, the possibilities are not promising. This country still has influence among Italians, but that influence has greatly diminished over the past decade. Italy now seems to be moving slowly and reluctantly toward a historic turn in its politics - and, for the first time in a generation, it is not waiting for Washington's approval.