I get a lot of questions about how to make Outlook remember the password when the Exchange account is configured using Outlook Anywhere.
When the computer is logged into the network, Outlook should log into the account automatically. If not, check the account settings in your Outlook profile. Go to File, Account settings, double click on the Exchange account then click More Settings and check the settings on the Security tab.
If the computer isn't logging into the network, or Outlook is not configured to always ask for credentials, you need to verify the credentials are being saved and your Internet Options settings (in Control panel or IE's Tools menu) allow automatic login.
Step 1: Verify the credentials are saved in the Windows user account. (Type Credential in the Start search field in Vista or Windows 7. Find the server address and click the expand button.)
Step 2: Add the Outlook anywhere URL to the Local Intranet zone in IE's Internet Option's dialog (Security tab).
Step 3: While on the Security tab, select Local Intranet zone and click Custom level. Scroll to the bottom and verify that IE is set to allow Automatic logon in Intranet zone.
If the computer is logged into the network, also check the settings on the Internet Options, Advanced tab. Look near the bottom of the settings list for Integrated Windows Authentication - is it enabled?
If you're having problems with the computer not remembering Sharepoint logons, add the URL to the Intranet zone and check that Automatic logon is enabled.