REPUBLICANS CONTINUE to look for a straight-ticket breakthrough in the South.

They tend to sweep the South in presidential races, and they have been electing Republican senators and governors in fair numbers for 20 years. Their problem is the offices farther down on the ballot. After years of effort, they still hold only 47 of the South's 133 U.S. House seats. They have only about one out of five seats in southern state legislatures. You won't find many Republican sheriffs or county judges outside a few anti-Confederate hill counties and some booming Sun Belt metropolitan areas.

That is true despite some impressive efforts. Recent elections show two potentially winning Republican strategies. One is to encourage what amounts to straight-ticket voting. They did this in 1984 in Texas and North Carolina, linking congressional and local Democrats not only to the Mondale-Ferraro ticket but, more importantly, to Democratic Senate nominees Lloyd Doggett and Jim Hunt. Mr. Doggett lost by a wide margin to Phil Gramm, and Mr. Hunt lost by only a hair to Jesse Helms. But for Republicans down the line, matching the Gramm and Helms showings produced a gain of five House seats in Texas and three in North Carolina -- more than half of the Republican net gain nationally.

The second strategy banks on nationally financed candidates, albeit with genuine local roots, campaigning as Reagan Republicans in traditionally Democratic rural areas. The prototype was Sen. Phil Gramm's candidate, former Texas A&M quarterback Edd Hargett, in the Texas 1st Congressional District special election last summer. Mr. Gramm hoped that a Hargett victory would inspire dozens of similar Republican candidacies in districts across the rural South.

But Mr. Hargett lost, though barely, and for 1986 the Republicans have few if any top-of-the- ticket candidates as strong and party-minded as Sens. Gramm and Helms. The Republicans will be making serious races in half a dozen or perhaps a dozen southern Democratic House seats. But the Democrats will be making about as many similarly serious challenges, including some in Texas and North Carolina districts the Republicans picked up in 1984. It should be added that Republican candidates are running serious gubernatorial races in Texas, Florida and South Carolina. But it's not clear yet that their strength will be reflected farther down on the ballot.

Nor are the national trends altogether heartening for Republicans. Party identification is sagging slightly in the polls; regional economic problems (oil, farming) hurt the party; the president's initiative on Nicaragua seems not very popular even in the South. So it looks as though the Republicans will be hard put to make significant gains, much less their long-hoped-fo breakthrough in southern congressional and legislative races. That must be frustrating for a party that has long sensed that on major national issues the South agrees with it. But so far this year the Republicans do not seem to have come up with a political strategy to persuade southern voters to abandon their traditional Democratic voting habits.