Banking? What bank's website does she use? If she doesn't trust FF, why can't you download other browsers? And can't you use other browsers for everything else, and IE for things which only work with IE?
Fine fine. Firefox is king, because of its numerous extensions and the development community that makes them. Until Chrome or Opera gets a library of add-ons rivaling FF's, Firefox is king.
Google Chrome is popular for its speed and simplicity of interface. In fact it is the fastest of all browsers. Opera and Safari are the other two popular browsers. I don't really have much to say about them. And there is also Konqueror and Maxthon.
So you can try out Chrome, Opera, Safari, Maxthon and Konqueror. I've use the first four.