Only allow access to the API for authenticated requests

private-wp-api.php

<?php

add_filter( 'rest_pre_dispatch', function() {
	if ( ! is_user_logged_in() ) {
		return new WP_Error( 'not-logged-in', 'API Requests are only supported for authenticated requests', array( 'status' => 401 ) );
	}
} );

I'd stress that if you want to lock down the API, you'll probably want to use a capability check rather than is_user_logged_in (such as current_user_can( 'edit_posts' )) as being logged in doesn't meant much if they are logged in as a user with no capabilities.