WHAT DO Herbert S. Walters, Elaine Edwards, Benjamin A. Smith II, Thomas M. Sorke, Sinclair Weeks and, now, Nicholas F. Brady have in common?

Well, each of them has been (or, in Mr. Brady's case, has just become) a U.S. senator. And if you haven't heard much about them lately--or ever-- there's a reason for that, too. Each was or is what is known in politics as a caretaker or, less politely, a seatwarmer--a senator appointed by a governor to fill a vacancy on the understanding that he or she will not run in the next election.

Appointing a seatwarmer can solve a sticky problem for a governor. For example, New Jersey's Gov. Thomas Kean, by selecting Mr. Brady, avoided the unpleasant task of choosing between other Republicans who are running for the seat in the June primary. An alternative is for the governor to appoint himself, which he can do by resigning on the understanding that his successor will sign the appropriate papers. The problem is that the voters do not take kindly to senators who have secured their own appointments. The last time a governor who had himself appointed won the subsequent Senate election was in 1942 (Happy Chandler of Kentucky); every one since has been defeated.

Not every interim appointee who turns out to serve only briefly begins as a seatwarmer. John Foster Dulles, who served in the Senate by appointment in 1949, lost a special election that year. William A. ("Dollar Bill") Blakley, appointed twice by Texas Gov. Price Daniel, saw one seat go to liberal Democrat Ralph Yarborough and lost the other to conservative Republican John Tower. Maryon Allen, appointed to the seat vacated by the death of her husband in 1978, surprised Alabama politicians first by running a spirited campaign for the seat and then by losing it in the primary. Muriel Humphrey, appointed to fill her husband's seat, considered running for the seat before deciding not to.

Seatwarmers often turn out to have been generous financial supporters of the governors who appoint them. That evidently is the case with New Jersey's Sen. Brady. He raised funds for Gov. Kean, as well as serving as managing partner of the Wall Street investment banking firm of Dillon, Read & Co. In a chamber full of professional politicians, with a couple of dozen members who daydream of being president, it is refreshing to have at least an occasional seatwarmer who looks forward to returning to Baton Rouge or Gloucester or even Wall Street. Welcome, Sen. Brady.