Just like windows, you will use a web browser which will handle the secure connection. Apple has their own, Safari. This uses the same secure protocols as other web browsers like Firefox & Internet Explorer
http://www.apple.com/safari/what-is.html#security - anti-spyware etc is included in the product.
Most banks will support Safari. Some don't e.g. Egg customers can log on to see their Egg accounts but Egg Money Manager, which lets you see accounts at other banks, only runs with Internet Explorer.