What happens if you don't pick-up the phone when it rings? (ask your friend to make a test call)
Does the computer modem still pick it up? (even if you don't - or does it only pick-up when you pickup?)
Does it always pick up after a set number of rings?
Other than your modem being defective and somehow being triggered by a ring-voltage on line, then all signs point to Fax software on PC causing it to pick-up calls.
Many people aren't aware that Windows XP has built-in Fax capability. It is possible your Windows XP fax software is triggering on incoming calls.
Check its settings:
Start-->Programs-->Accessories-->
-->Communications-->FAX-->Fax Console
Then, in Fax Console window, Check settings:
Tools-->Configure Fax (config wizard setup).
Make sure Send and Receive Faxes is "Disabled".
If this is not the cause of pick-ups, then just disconnect phone line from PC when not using for Internet browsing. Then, there is no way PC can interfere with incoming phone calls. Reconnect line to PC only when you want to surf.