Have a System DSN connecting to an SQL server with the username and password saved in it but the user is still prompted for them the first time they connect to the database

To NOT be prompted, Setup user accounts and passwords in SQL that match the NT logins. Then start the "ODBC Data Source Administrator" and select the DSN and click 'Configure' , click next, select the "With Windows NT Authentication using the Network loginID " option and then click Next, and then Finish.