You ask if you need antivirus software on you desktop. My answer is not about technical issues, it is about legal ones. Where I live, banks require you to have antivirus software installed on the computer which you use for online banking. If you have antivirus (and some other measures), there's a good chance that you get your money back. It depends on many things, and there is no guarantee, but they actually do this. From their own logs they can see many things and they see patterns that show that the account really is hacked.
If you don't have antivirus, and your account gets hacked, they can blame you and they might refuse to refund your money. Then you can go to court and prove them wrong, but are you willing to take that risk?