If OneCare is working on that PC in all other regards, the problem isn't with your OS version, since the trial installed fine.
It would appear that something is otherwise interfering with the activation process, which needs to launch a browser session and connect to the Internet for you to log in with the subscription LiveID. Is IE your default web browser?
If you open OneCare, do other links in the main interface work fine such as Help, Change Settings, Your OneCare Circle, etc.?
The "experienced a problem and must close" leads me to suspect a larger problem with your OneCare installation, but would find it strange if only activation was impacted. How long has the trial copy been installed on the PC?
Have you considered reinstalling OneCare to see if you can proceed to activation?
-steve