I am running a SharePoint Foundation 2010 site which uses claims based authentication. (https://subdomain.domain.com). This is done using a Secure Token Provider. The authentication/authorization works very well for the entire site except documents. Users are asked every time to re-enter credentials. I have seen some examples that users would need to add this site to the local trusted sites in IE. Is there another method? A good amount of my users are not too tech savy and would probably struggle with this task. I also tried to select no for "Enable Client Integration" on the site.
This it problem as Word (or any other program) does not know how is user logged in within IE. The only thing they can do is to enter credentials and check Remember credentials. This stores entered credentials to local user store and allows Word to use them next time. But as you mentioned "Remember my credential" functionality does not work all the time. To be sure Trusted sites is good way, but HTTPS connection should also allow users to remember credentials. Trusted sited as well as others IE settings can be set via group policies.