Commonwealth Bank? I'm not sure what you mean by "government bank". Is the bank in the US?
I'm serious about changing banks. If that is not an option for you then the government ought to buy you a government computer for use with the government bank and absolve you of any responsibility for its use or misuse.
Browsers are either compliant with internationally established security standards, or they're not, and both Chrome and Internet Explorer have been hacked.
http://www.infoworld.com/d/security/researchers-hack-internet-explorer-11-and-chrome-mobil e-pwn2own-230854
NO version of IE is safe: https://technet.microsoft.com/en-us/library/security/2963983.aspx
Any bank that requires any particular browser should not be trusted with your money, because they do not understand the most fundamental aspects of information security.
Safari and OS X 10.7.5 are as secure from threats and "hacking" as anything available today. Anything better does not exist.