If a bank website is hyper-secure, what might data gathering viruses see if I typed in passwords to access my account online?
If the bank's website is secure, and if your own computer is virus-free, they will exchange data over an encrypted connection. (If a web address begins with "https://", that means the connection is encrypted.) A virus in the middle will pick up the encrypted gibberish.
In principle, attackers can decrypt this gibberish. But in practice they won't, because decryption is expensive in computing power. From their perspective, the best line of attack isn't against the bank's computer or the connection. It's to get on your hard drive.
Once the attackers have a virus on your hard drive, they can monitor your keystrokes and read your screen. Whatever you know about your bank account, they can find out, too. So the key to securing your bank transactions is to keep your computer virus free. If you manage to do that, and if your bank exchanges account information through encrypted web pages, you're fine. (I know Ebay and PayPal do encrypt their transactions.)