It looks like your doctor is starting conservatively.
Side effects of steroids include increased drinking, increased
urination, panting, weight gain, and increased susceptibility to secondary infections. If you feel that your dog is having trouble breathing, then he/she should be seen. If it is just increased panting, it is a side effect of the drug and not something that would make the doctor take he/she off the drug unless the dog was excessively agitated. I hope this answered your question.