Safety you can bank on: Chromebook, Linux, phone
If you're not deterred by learning strange software, you can save hundreds of dollars by downloading a copy of the open-source Linux operating system and burning it to a CD or copying it to a flash drive. As security journalist Brian Krebs explained in the summer of 2012, you can pop that into your Windows PC, boot the machine off it, and go online insulated from whatever might lurk in your copy of Windows.
(In that post, Krebs endorsed a version of Linux with the charming name Puppy Linux; I usually recommend a different variety called Ubuntu, but the differences don't amount to much in this context.)
Using Linux just for online banking also insulates you from most of its potential complexity: You're only running a browser.
But if installing new apps in Windows already fills you with dread, or the thought of picking one version of Linux out of dozens makes your head hurt, spend money instead of time. A Chromebook just might work — and might be all the computer you needed in the first place.