Security has more to do with the user
(you) then the software. Windows can be used securely and Linux can be used insecurely, for example. Furthermore it has a lot to do with your bank. Ubuntu cannot protect you if your bank has predatory policies, whereas if your bank has a fraud department and actively protects you, that's better security than any operating system/browser can provide. So get on the phone and call your bank and ask THEM "what are you doing to protect me from online fraud?" not some strangers on an internet forum!
That being said, here is a good introduction to security in Ubuntu, and all topics apply equally to Ubuntu Studio (which is simply Ubuntu with preinstalled audio/video applications):
Personally, I do online banking all the time in Linux. I use my Firefox's Private Browsing feature, though I won't insist that is strictly necessary.