Windows Auth is usually the preferred method of authentication. The prompting will occur on any SP site over the internet. I have never seen the Save Credentials box actually work.

If the SP site is moved internally and accessed using Active Directory credentials then the prompting will stop, if the client is using the same Active Directory setup. (e.g. user logs in to their computer with their domain credentials)

There may be other ways to suppress the prompting with single sign-on, but moving internally (access via VPN or other sources), is one way to prevent the prompting.

There is no way that I have found either, and I agree with Carsten. You may be able to use the browser to save the credentials, but that doesn't always work either.

I have a client for whom I set up a set of calendars for conference scheduling, and I told her to have each of her users log into their calendar and do a "Connect to Outlook". She decided to perform that action on a bunch of her users' calendars without asking me. She couldn't figure out why, when firing up Outlook, that she got prompted 14 times for a password. Well, she had 14 other users' calendars synced to her Outlook.