This is not likely to be related to malware on a Mac. Unfortunately, many bank policies don't take the type of computer you're using into consideration. They probably will not re-enable your remote access just because you're using a Mac, so you can either lie to them about having installed anti-virus software or you can install one of the free options from the App Store. VirusBarrier Express and Dr. Web Light are both good options. They cannot destabilize your Mac as a whole, because they can't install any code that runs when they're not running, and won't slow you down except when they're open and running a scan. Once you have run a scan to satisfy the bank, you could just leave the app sitting inert in the Applications folder and honestly say that you have anti-virus software "installed."
Note that, if you actually find any malware, it's almost certainly going to be Windows malware that has gotten onto your hard drive somehow, perhaps by being attached to e-mail. Post the names of anything found here and we can tell you whether you need to worry about it or not.
As for general methods for preventing this in the future, Linc has some very good suggestions. I'd also add that you need to do the same with the e-mail address associated with your bank account. You don't want someone hacking your e-mail and intercepting bank notices. Also, watch out for phishing attempts... if you get an e-mail from your bank asking you to log in to your account for any reason, don't click any of the links in that e-mail! Instead, log on to your bank site as you normally would do, using a bookmark in your web browser or the like.