This doesn't sound like a problem with konqueror; it seems like KDE has been installed improperly.
If you haven't done so already, it is always good idea to log off/log back on after updating KDE.
If it is a problem with konqueror itself, this will tell us what is going wrong:
Open konsole. Type:
..then press enter. Konqueror will open. Browse the internet until it crashes. Inside konsole will be a log of all the errors that happened. Paste them here.
It would also help to know exactly which websites konqueror is crashing on.
It is much more likely that you just have to reinstall KDE. To do this, open yast, then under software, choose Software Management. Search for kdebase3. Right click on kdebase3, and choose update. Finally, click accept.
System Update, located under software in yast is another option you may want to explore.
Or, as another poster suggested, upgrade to SuSE 10. KDE 3.1 is somewhat outdated. Upgrading will certainly fix your problems
OpenSuse has a 10.0 "evaluation version" available for free. (It won't expire or anything)