STABILIZING THE American dollar is going to take more than cheery handshakes and optimistic communique's at the financial meetings now getting under way here. Amicable meetings are certainly better than the other kind. But preventing a further fall in the dollar depends, first of all, on interest rates. The American economy is now being fueled by a heavy flow of investment from the rest of the world. Without that flow, the dollar sinks like a stone. To keep the foreign money flowing in requires interest rates here that are higher than those abroad. To preserve that essential stream of foreign money, American interest rates have been rising.

Those investors are betting not only on current returns but on the future prospects of this country. That's why it was crucial that President Reagan agree to sign the bill that puts the bite back into the Gramm-Rudman-Hollings machinery, Congress' device to force down the federal budget deficit. He complained bitterly about having to sign the bill, but the important thing is that he will do it. If he had refused, the financial world -- much of which is now gathered here in Washington -- would have concluded reasonably enough that the United States was giving up the fight against the deficit.

This week's gathering, the annual meetings of the World Bank and the International Monetary Fund, is the one occasion on which most of the world's governments come together to talk about the economic policies on which their common prosperity depends. The pressure on the exchange rates, one of this year's key subjects, arises from the United States' inordinate need for foreign money.

Supporting American society in the manner to which it has become accustomed currently requires nearly $150 billion a year of foreign support. That need arises from the very low American savings rate, which produces not nearly enough capital to finance both private investment here and public deficits. That requirement is beginning to decline, but only very slowly. Meanwhile, private investors abroad have begun to back away from the dollar. Without their contributions, it is unlikely that governments alone can keep exchange rates at their present levels for long.

The weakness of the dollar is not a matter of technical esoterica, comprehensible only to the experts. It is a reflection of the way Americans, public and private, have come to spend -- and to depend on borrowing to keep spending. That has now become a threat not only to their own economy but the world's. Beneath all the official good cheer at this year's meetings runs more than the customary undercurrent of anxiety.