Definitely. Although there is no such thing as perfect security, a properly configured Linux, *BSD or Unix system can be very secure and orders of magnitude better than any version of windows.
Linux, Unix and *BSD are 100% immune to windows viruses and exploits as long as you don't install WINE ( WINE allows windows software to run on Linux ). There are about 80,000-100,000 windows viruses, while Linux only has about 100, MacOS X 60-100 and Unix about 10-100.
If you use well written firewall rules, SELinux and you configure and use the system properly it can be very hard for anyone to break your security.
Some security experts recommend using only one computer or a secure Virtual Machine (VM) running on top of Linux for your banking...