According to Bon Appetit, it is completely safe to drink pickle juice. The online magazine recommends drinking the juice as one way to deal with the liquid left over in a jar after all the pickles have been eaten.
The New York Times reports that, for some people, pickle juice is even beneficial. The Times relates a study from Brigham Young University that investigated giving pickle juice to exercisers who had become mildly dehydrated. When a toe cramp was induced, the exercisers who drank pickle juice recovered far more quickly than those who drank water or nothing, according to the Times. Researchers speculated that the juice was sending a nerve signal that relaxed the cramped muscle.