It can be beneficial to eat both before and after a workout. However, whether a person eats before or after a workout, or both, is not as important as what he eats.
For a light workout, it is fine to not eat anything beforehand. For a more rigorous workout routine, quality carbs and lean protein are the best foods to eat. It is also important to eat foods that are low in fat and low in fiber before a workout. After a workout, foods that are high in protein are the best to consume, as protein helps muscles recover and grow after exercising.