The best way to find a healthy dog to adopt is to visit a trustworthy rescue group, such as the Humane Society. Good rescues have a veterinarian examine the dogs for any health problems before making them available for adoption.
Most large cities have many rescue groups, so spend some time researching rescues in the area. To make sure the one you're considering is good, ask about its health and behavioral testing for dogs. Make sure the dogs come microchipped, vaccinated, and spayed or neutered, and that the rescue is responsive to questions. A good rescue also screens adopters to make sure they can provide adequate care, so expect to be interviewed and fill out an application.
When you meet the dogs up for adoption, examine their living conditions. The shelter or foster facility should be clean and well-maintained. Watch for dogs that seem to be wheezing or coughing; excessive mucus in the eyes or nose can also be signs of illness. Shelter dogs are usually not all in top condition, but be alert for signs of excessive hair loss or skin irritation that could indicate ringworm, fleas or mites. If you do notice any signs of potential illness, ask the rescue staff about it.