How can I check whether my AirPods are fake or genuine?

If you bought them from an Apple store or Apple-approved store then you can be sure they are genuine.

Otherwise there is no way to tell a good fake. The fake makers can also fake genuine serial numbers, so online checks are no guarantee. You could put a call in to your local Apple Genius Bar and ask if they would be willing to authenticate them but it might cost you and they might damage them in checking.