>> "She got emailed by her bank that using Windows XP would not be safe anymore"
Before switching to Linux I'd recommend checking the bank supports it. I know a couple of banks in my country that use weird security systems that won't run on Linux/Mac.
Assuming there aren't any plugins or applets involved in the authentication process. Sweden, for example, is moving to a system where you use a smart card and USB card reader for authentication on most/all bank and government sites.
The smart-card based banking systems I am aware of don't run in the browser (not saying they couldn't have chosen to work via a plugin, they just don't seem to). Gnucash has for several smart card systems, and at least HBCI/Fints support is good [1,2].
In Belgium, they used to bundle a java app with some certificates or keys on a floppy disk years ago. Fortunately most of them have switched to a custom calculator-like device that spits out verification numbers and doesn't require any connection to the computer or the network.
Before switching to Linux I'd recommend checking the bank supports it. I know a couple of banks in my country that use weird security systems that won't run on Linux/Mac.