A noisy unlatched gate can cause quite a nuisance, especially in the path of a forceful wind. That nuisance can lead to bigger issues if the fence belongs to a neighbour. You need to approach the situation in a delicate manner in order to preserve the good will between you and your neighbour and ultimately get him to latch the banging gate.
Think of a clear idea of what you want to say to the neighbour. Consider what result you want, and then define the actions necessary to achieve it.
Write an outline of the conversation you intend to have with your neighbour. Keep the tone friendly, as if you're in need of a neighbourly favour. This will keep the neighbour from becoming defensive and allow him to be more receptive to your request.
Visit your neighbour in person. Try to choose a time when the neighbour is likely to be available, such as after work or while she is in her yard.
Greet your neighbour and start the conversation with a compliment. An example approach would be, "Hi, Jamie. I was hoping I would catch you at home. Those shrubs you planted last spring have really filled out nicely."
This will allow the neighbour to lower her defences and hopefully cause her to be more receptive to your forthcoming request.
Address the concerns you have about the noise caused from the gate banging. Explain that you were hoping there might be some way to keep the latch closed in order to prevent the banging.
Mention that keeping the gate latched will prevent damage to their property by eliminating the banging occurrence. By adding a benefit to the neighbour, you make him feel like he is also gaining something for his effort.
Finish your visit by asking the neighbour if there is anything you can do for her. It's usually a good idea to close communications on a friendly note. Instead of resenting you she will likely think of you as a good neighbour, which is something most of us hope to attain.
Be careful that your facial expressions coincide with the friendly tone of your conversation. Body language can sometimes speak louder than words.