WILL THE SOUTH dominate the presidential selection process in 1988? A lot of southern politicians hope so. Govs. Bob Graham (D- Fla.) and Mark White (D-Tex.) and southern state legislators meeting in Stateline, Nev., have urged that all or most of the southern states hold their presidential primaries, caucuses or conventions on Tuesday, March 8, 1988, or Saturday, March 12.

Don't mark your calendars yet. Though this is allowed under both parties' rules, some 15 state legislatures have to act in tandem to make it a reality. There is genuine enthusiasm for the proposal. But before lobbying 15 legislatures, its proponents ought to think hard about how it would work, and consider a modification.

The impetus for the southern primary comes from Democrats in the region who resent the influence Iowa and New Hampshire have currently. They think Iowa is too dovish, New Hampshire too quirky, and that the combination, together with the early activity of such groups as the AFL-CIO, NOW, and gay rights organizations, tends to produce Democratic nominees unpalatable to the South and unable to win a general election. A 15- state southern contest, they argue, coming at the beginning of the process, would not only take up most of the candidates' time but also dominate their decisions on strategy, thus producing a nominee more in conformity with national opinion.

There is something to that. The South is, in important political respects, different from the rest of the nation. But those differences are muted in Democratic primaries, which, thanks to the South's Democratic tradition, take in the large majority of the electorate. Thus Democratic contests in the South give you a closer replica of the November electorate than contests elsewhere.

But holding 15 primaries and caucuses just a week or 10 days after New Hampshire's still leaves a risk of the tail wagging the dog -- of the huge bloc of southern votes being swayed by the momentum from the North. Recall that in 1984 Gary Hart, not a household word in the South or a politician who speaks in southern timbre, zoomed from 3 to 35 percent of the vote in Georgia within days of his upset victory in New Hampshire.

If the promoters of the southern primary don't want to see something like this multiplied by 15, they could try to persuade their legislatures to space southern contests out over three weeks. That would give most voters in the South time to digest results not only from the North but from their own region, and it would hold the nation's political attention on the region not for one week but three. The spotlight may be shared, notably with Illinois, whose March primary was perhaps decisive in 1984 and 1980. But the extended opportunity for reflection and examination of the candidates may more reliably achieve the goals of making the South influential and getting the Democrats a stronger candidate.