How can I change DNS providers without needing to type in my password?
The account you are logged in as needs to be configured as an admin account, because the DNS settings affect all users on the system, not just the current user.  If it is not an admin account, then you will need to authenticate every time you want to change the DNS settings.

If you are logged in using admin account and still have the problem, then open System Preferences, go to Security &amp; Privacy, click the "Advanced..." button, and make sure that "Require an administrator password to access system-wide preferences" is not enabled.