WHAT IS the potential of the Christian evangelical movement in politics? Greater than you might think, to judge from news in Washington. On Capitol Hill and in the halls of the executive branch the evangelicals have not been particularly successful in achieving or even clearly articulating their goals. But in Iowa -- not heretofore considered a center of the evangelical movement -- they have achieved some significant local political successes, with implications for 1988 and beyond.

The successes came in February's Republican precinct caucuses, where well-organized evangelicals won majorities or sizable minorities. And not just in scattered rural precincts, but also in counties that include Des Moines, Cedar Rapids, Sioux Falls and the university town of Ames. Evangelicals are organizing as well in the counties around Grand Rapids, Mich., where the precinct delegates elected this August will control the choice of presidential convention delegates in 1988.

The expectation among Republican leaders is that evangelicals will support TV evangelist Pat Robertson's expected presidential candidacy two years hence. The more interesting question, in the meantime at least, is: what do these new recruits to active politics want now? The platforms they passed in last month's Iowa county conventions talk, as you might expect, about the need to prohibit abortions and ban pornography. They support the Reagan administration's Strategic Defense Initiative and aid to the Nicaraguan contras -- policies stressed by a lot of national conservatives, who note that they hold together otherwise diverse conservative followings. They hark back to the old days of Proposition 13 and Kemp-Roth by opposing increases in local property, state and federal taxes. All pretty standard stuff in American politics these days.

But they also sound a plaintive note. "Rural America, where is it going?" one county's Republicans ask -- a poignant question given Iowa's troubled farm economy -- and call for lower interest rates. They argue strenuously for parents' rights not to send their children to school but to educate them at home, where they won't be exposed to "values clarification" but can be required to say prayers and to pledge allegiance to the flag.

Such feelings, multiplied many times across the country, give the evangelicals the potential for being a national political movement, with an important effect on the Republican Party and perhaps on national policies. Some people fear that an evangelistic horde will force its values down the throats of other citizens. But so far the evangelicals seem to be engaged in rather conventional politics, backing policies already widely advocated or concentrating on issues that are ordinarily the stuff of local school board elections. For all their discontent and rhetoric, their tactics and goals, like those of insurgents in party politics in the past, may be more in line with American political traditions than they or their adversaries may think.