Prednisone is prescribed when the dog's immune system is attacking itself, causing inflammation. Since diarrhea can be caused by intestinal inflamation (see causes of diarrhea) a corticosteroid such as prednisone (otherwise known as steriods) is prescribed in addition to other approaches such as alterning your dog's diet. When my dog was diagnosed I was told that he had IBD or Inflammatory Bowel Disease. Or said another way, what a mess all over the house (what we do for our dog).
In addition to helping stop or slow the diarrhea, steriods also help deal with the dehydration by helping your dog absorb more of the water he drinks and as in my case, my dog started to regain his interest in eating (like father like dog).
Prednisone is a good news, bad news kind of drug. The bad news is that it is not a cure and will probably be part of your dog's life for years to come. The good news is that with a relatively low dosage, the diarrhea should be manageable (or at least before you can't clean another dog doo off the floor).
Be sure to work with your Veterinarian to monitor your dog's ongoing health.
Dog Prednisone and Diet
A common byproduct of having IBD or diarrhea due to IBD is that your dog may need some nutritional supplements. Start with what should be eliminated, which includes anything that is left over on the family's dinner table.
See our discussion on diet on how to find the right combination of foods for your dog. It worked for mine (although keeping him away from what he shouldn't eat such as grass and bugs is still a challenge (anyone know a good dog psychologist?).