NONSENSE! if all you are doing online is telling your bank through a well secured bank site (and by this I mean your bank must text you a one time password via your phone before you can access their site) where your bill pay is pre set up. You have very little security issues to worry about as the only thing "visible" is the nickname you have assigned to the accounts you want paid. All the information that is at risk resides in the bank. You are only passing a message of "pay MY NOrdstom acccount this much $$. Nothing that is vulnerable to hacking.
You can safely do this via a public network. What you shouldnt do on a public network is give out your CC numbers in full unencrypted.
Using your usual Bank web site to pay bills is not a problem.