When we uncheck "Admin Password is required" option in Management console -> environment->advanced_settings->webservices it gives an error.
Uncheck Admin password is required and save.
Error when trying to save by unchecking "Admin Password is required"
By default, Admin Password Required is set to true (checked).
The message "At least one type of authentication is required" is correct and expected. You really do need to have one of the authentication methods selected.
If you don't want to provide the admin password in your request you'll need to use either the WSS Token or SiteMinder Authentication.