The highest version of Safari you can run under Panther 10.3.9 is Safari 1.3.2---pretty old considering the current version is something like 5.1.7.
You can run a browser that, although not necessary current, has been updated since Safari 1.3.2:
This version of Camino runs under Panther: Camino 1.6.11
iCab 4.9 also runs under Panther: http://icab.de/dl.php
If you post the PowerBook's screen size (12, 15, or 17-inch), the processor speed, and amount of RAM you have installed, we can see if you have something worth updating and how to do it.
would you recomened download the next O/S after panther?
No. There is no legal download of any Mac OS version higher than 7.5. Downloads that are offered are illegal and may damage your computer. The next version, OS 10.4 "Tiger" is in high demand and a proper full-retail install disk is not longer available from Apple and can cost more than some used PowerBooks are worth today.