How to check if the user registration is allowed/active?

You can check it very easily with the help of get_option.

Here is a simple code for checking if user registration is allowed.

if ( get_option( 'users_can_register' ) ) {

    // Your custom code or message to display if user registration is allowed.

}

get_option( 'users_can_register' ) returns boolean true (1) or false (0).