Final answer:
PGP-encrypted emails would be the best communication method to recommend.
Step-by-step explanation:
The BEST communication method to recommend in this scenario would be PGP-encrypted emails. PGP (Pretty Good Privacy) is a cryptographic protocol that allows for secure communication by encrypting the email content. When using PGP-encrypted emails, the email servers and clients are encrypted, ensuring that the content remains secure during transmission. This method meets both the criteria of encryption on the email servers and clients and being safe to transmit over unsecured Internet connections.