I presume that the OP is located in the US and using US-based banks. If so, let me tell you ALL US based banks have pathetic security. So no matter what you do, you will never know where the information will leak. Someone can sniff through your wi-fi and what not.
I say this because the way the log in systems are set up. Some banks have few security questions, some banks have cookies and I have never understood how they call it secure. When I was living in Europe, their cards have SIM like chips (10 years ago) so your ATM transactions are secure, even your store purchases are secure (Nobody can steal your card info by copying or RFID).
The bank log in have one time codes printed on a sheet and you have to use certain code only (they provide the hint) or you had a tiny gadget that will generate code for you or the gadget you need to hook up to your machine through USB and stick your card in the gadget to log in. I do not see anything in the US. When asked, they tell me you are covered if fraud takes place.
So I use Chrome.
Before I use bank, I close all browsers, I run CCLeaner, PrivaZer and restart the machine. I log into Mint, update all accounts and see if everything is fine. Again, close all browsers, run all the stuff mentioned previously and then log into the bank account.