Securing ESP8266 connections
The simplest method is to encrypt your communication channel using some secret key.  You could use the built-in AES encryption functionality, which would mean that only people that know the key could encrypt a request in such a way that the device at the other end could decrypt it.

You would think that SSL (HTTPS) would do the job, but that only provides interception security, not authentication. Of course for even more security you can combine both HTTPS and AES encryption.  One for overall encryption, and the other for encryption-based authentication.