There is no 100% for sure way to determine authenticity when purchased from a third party. The counterfeit market for AirPods is huge and all materials are easily faked and the serial number is just a copy from a legitimate pair, so no serial number checks will be useful. You may experience these problem with fake AirPods:
- Poor Audio Quality
- Noice Cancellation may not work
- Find My may not work
- Poor Bluetooth Range
- Connection problems
- No AppleCare support
If you want to be sure you have a legitimate pair of AirPods, your only option is to purchase from Apple or Authorized Apple Seller. Many times when purchased from third parties, you will find an original box that contains AirPods that have already been paired to another device and returning them is not likely. It's a risky move.