This is a very common side effect with antibiotics, so is vomiting. Encourage liquids (water or diluted juice in a sippy cup) and monitor for dehydration, although that is rare. I'm a nurse and this is what I tell parents when their child is given an antibiotic. The medicine is destroying the bacteria in her ears that is causing the infection, but is also killing the normal, healthy bacteria in her GI tract that regulates digestion.
I forgot to add that you can give whole milk yogurt to your baby and that may help with the diarrhea.