WHO NEEDS a lieutenant governor? The question is prompted by the nomination by Illinois Democrats of a follower of Lyndon LaRouche for that office. That has put gubernatorial nominee Adlai Stevenson III in the uncomfortable position of having to extricate himself somehow from his ticket-mate, because in Illinois, as in most states, when you vote for governor you also vote automatically for his party's nominee for the second slot; and no sensible voter wants Mr. Stevenson's running mate to be a heartbeat away from . . . well, the command of the Illinois National Guard or something like that.

This absurd situation raises questions about the competence of many Illinois politicians and voters. But it also raises the question: Why have a lieutenant governor at all? As we write, some 46 million Americans get along quite nicely without one. New York's Lt. Gov. Alfred Del Bello resigned last year, not to be replaced until this fall's election; Ohio's Lt. Gov. Myrl Shoemaker died in mid-term; Arizona, Maine, New Hampshire, New Jersey, Oregon, West Virginia and Wyoming have no lieutenant governors at all.

But don't you need someone to succeed the governor if he dies or resigns? Well, each of these states has a designated successor who has more important duties and responsibilities than all but a handful of lieutenant governors. True, some southern states -- Virginia, North Carolina, Texas -- make the lieutenant governor the real operating head of the state Senate, and occasionally a longtime incumbent, such as William Hobby of Texas, becomes a highly useful and constructive official of state. But usually the office of lieutenant governor is as bereft of useful work as the vice presidency ordinarily has been, at least until Presidents Carter and Reagan gave Vice Presidents Mondale and Bush important assignments and access to the Oval Office.

Yet for all the evidence that the lieutenant governorship is the vermiform appendix of most state governments, the office has been growing more popular. Maryland elected its first lieutenant governor in 1974 (and a man of high qualities, the late Blair Lee III); New Jersey is talking now of getting one; the newest states, Alaska and Hawaii, changed the title of their secretary of state to lieutenant governor. You can understand why incumbents Stephen McAlpine and John Waihee III must prefer to be addressed the rest of their lives as "Governor" (you definitely don't call the poor officeholder "lieutenant") rather than "Mr. Secretary." But it's harder to understand why so many states think that the way to have a worthy successor on hand for their governor is to hire a man or woman for a job with little or nothing to do.

Nor is it apparent why more and more states have made the candidates for lieutenant governor and governor a kind of package. Yes, it's anomalous to have a governor of one party and a successor and substitute of another. But presumably the voters who chose the two candidates thought that combination was the best choice. California, Idaho and Iowa live with such divisions now; Virginia did a few years ago. Sen. Paul Simon (D-Ill.) served as lieutenant governor under Gov. Richard Ogilvie (R) in 1969-72, with results certainly less ludicrous than the situtaion in Illinois now. Is this office really necessary?