A dog's sneezing and runny nose may be caused by a cold or canine distemper, because the early symptoms of canine distemper are similar to those of a cold. If the dog shows other abnormal symptoms, such as vomiting and diarrhea, blood in the stool, and not eating or drinking, It is best to take your dog to a pet hospital for examination first to confirm the cause and then treat it. If you have a cold, you can give your dog special cold medicine for pets, and it will get better in about a week.