The type of cold a dog suffers from is different from the type a human suffers from. The illness is not communicable between species — at least, one has not yet been discovered — so there is no need to worry about catching your dog’s cold, or vice versa.
If your dog is coughing or sneezing, but is in otherwise good health, you may be able to treat the condition as you would a simple human cold — with lots of liquids, healthy foods (Chicken soup, even? But of course! Just make sure to leave out the bones.), warmth, and maybe even some time in a warm and humid room. This can be done by placing a humidifier near his rest area or by filling the bathtub with steaming water and letting the dog hang out in the bathroom for a bit (not in the water), just to let the steam loosen up his sinuses and lungs.
If your dog is either very young or very old, it is best to have her looked over by your veterinarian, since dogs at either end of the age scale tend to have less capable immune systems and can suffer more as a result, and if symptoms don’t improve or appear to worsen, consult with your veterinarian.
You can help to prevent a cold by keeping her indoors during cold, wet weather, with just brief trips outside for relief. It isn’t the cold temperature that creates the illness, of course, but over exposure to unfriendly temperatures or environments can create a physical situation that makes it easier for a bacterial or viral germ to latch on and take hold in the body.