THE RADICAL students who started the riots in South Korea are being joined by some older people of the middle class and cheered by still others. A government whose pride is order has lost control of the streets. The military shows signs of unease, with one faction growling for a harsher crackdown, perhaps martial law, while another explores a measure of relaxation and a return to the talks on presidential-election rules whose breaking off, by the government, triggered the crisis. A return would raise the question of whether the opposition would or could control the charging bands in the streets.

Caught unprepared, the United States struggles to retain a policy grip. It had hoped that President Chun Doo Hwan would guide South Korea to its first peaceful transfer of power next February and that Seoul would reap the prestige of hosting the 1988 Summer Olympics. But Washington did not count on President Chun's rigidity or on the determination of the opposition. Now the administration must play catch-up: Mr. Reagan has written Mr. Chun advising moderation, and endorsed Rep. Stephen Solarz's resolution calling for broader freedoms and the resumption of constitutional talks.

There is no text to guide a transition toward democracy in a country such as South Korea, which has no firm democratic tradition and which has a treacherous neighbor, communist North Korea, and the powerful armed forces to match. These cultural and circumstantial considerations have reinforced the caution of American policy makers in dealing with violence or revolution; they fear decisive action for democratic change may offend nationalistic sensibilities or embolden the players to do the wrong thing. The Philippines, where the Reagan administration did help move events along, is cited as a relevant model. But the Philippines had a democratic strand in its past and no comparable security threat. Crucially, a plausible democratic leader came along and the army suddenly switched to her side.

There are, nonetheless, risks in the stated American policy of ''restraint.'' South Korea is a place where the society, modernizing fast, has outgrown the state: it is readier for democracy than the generals are. It is also a place that hangs on every American word -- and on every American hesitation and silence, which some in the opposition exploit to blame Americans for the home-grown repression. The task before the United States is to make sure Koreans understand we are on the democratic side.