It depends on your goal- if your primary goal is fat loss, you may would only want to eat the banana post-workout, if at all.If your primary goal is just general fitness, or muscle/strength gain, by all means eat the banana both before and after!
Personally, I always take a really ripe banana with my protein shake post-workout for the following reasons:
1.) Bananas are an excellent recovery/energy food- they contain certain minerals that restore and enhance muscle function.
2.) The natural sugars in bananas are perfect for restoring optimal glycogen levels in the muscles, thereby restoring an anabolic state within the body (exercise initiates a catabolic response (muscle break- down) ) Also, the sugars will digest and cause a quick insulin response, providing the perfect "shuttle" for quickly assimilated proteins. Quick delivery is so vital because the body's ability to absorb glycogen and protein is elevated only for a short time post-workout.
So yes, bananas if taken within an hour following your workout will help you recover faster. Don't forget the protein! (A fast-digesting whey is ideal)
You could also eat a banana as I mentioned before your workout as well for extra energy and stamina. Be sure to eat a complex carb source pre-workout as well with your bananna and be sure to include protein and a moderate amount of fat!