Dogs can eat bananas, but should only eat these or any fruits in moderation. A dog's primary diet is meat and protein. Dogs do not need fruit, and they should not be given any fruits with a lot of citric acid such as oranges or lemons.
When deciding what to feed your dog, a rule of thumb is to stay as close as possible to the dog's natural diet. A dog in the wild would not normally seek out fruit to eat. If your goal is a healthier dog, then it is a better choice to increase outdoor activities and exercise and feed your dog high-quality protein.