I have used a bank that focuses on internet customers for 15 years, American Bank.
https://ambk.com
They reimburse ATM fees and the checking accounts are interest accounts, albeit small interest right now.
In 15 years, I have never had any issues with them. I can pick up the phone and talk to the woman who helped me originally set up my account 15 years ago. As long as you are willing to not have a branch locally, I highly recommend them.
Log-in credentials use a SecureKey for the password, which means that the password changes every 30-40 seconds. I would think it is hard for criminals to crack.