And you'll be seeing that message more frequently if you're really still using OS X 10.3. You can try another browser such as Firefox (and upgrade that browser as far as you can), but, truth to tell, more and more sites are demanding that you use a browser written in the past few years. If you're going to stay with OS X 10.3, you could try Classilla (http://www.floodgap.com/software/classilla/) instead of Safari, but I have no idea how well or even if it's going to work with most sites today.
If you move up to OS X 10.4, you could use TenFourFox (http://www.floodgap.com/software/tenfourfox/) as your browser. How long the TenFourFox project will continue to be updated is anyone's guess.