I’d like to sharing a solution of this issue on SharePoint 2007. 

User was prompted for his/her password when they open document library with Windows Explorer even though he/she was already logged in at the customer portal.

I found the solution in Microsoft KB. Almighty MS KB!

http://support.microsoft.com/KB/943280

It’s a client issue, not a Server issue.

Here is the step for client:

1.Click Start, type regedit in the Start Search box, and then press ENTER.

2.Locate and then click the following registry subkey:

HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Services\WebClient\Parameters

3.Find the AuthForwardServerList and select it.

4.On the Edit menu, click Modify.

5.In the Value data box, add the URL of the https://****** or http://******, and then click OK.

It seems like adding the site to trust site in IE settings. But I have let the user try this method (add the url to trust site in IE settings). It doesn’t help. You can see the root cause in that MS KB.