Writers talk about what influences their work; In Praise of What Persists, edited by Stephen Berg. New York: Harper & Row. 289 pp. $14.95.

Editor Stephen Berg had the good idea of asking 24 contemporary American writers for essays describing the major influences on their work. What makes the results so interesting and illuminating is their runaway variety.

Many tip their hats to other writers. Cynthia Ozick grudgingly admits her debt to Henry James, while warning young writers to ''keep . . . psychological distance from the supreme artists.'' Robert Coles cites the King James Bible and William Carlos Williams; Reynolds Price acknowledges his dependence on Hemingway; Robert Coover pays homage to the unlikely tandem of ''Don Quixote'' and Samuel Beckett; Leonard Michaels offers extravagant praise of Chekhov.

There are remembrances of favorite teachers and of stimulating family situations. Poet Tess Gallagher finds worlds of meaning in ''My Father's Love Letters''; Grace Paley muses over the contortions of aesthetic choice and the need to render reality honestly.

Some of the essays are unsparingly personal: Raymond Carver confesses his obsession with the ''unrelieved responsibility and permanent distraction'' of being a family man; and Gerald Stern finds in ''my own loss and my own failure'' his most compelling literary subject.

A fine, provocative collection that should be of interest to all writers and readers.