The accepted answer on WordPress stack exchange was not the one I went with.
Just in case:
My implementation was like this:
<?php
// 'menu-header' is the value of 'theme_location'
if ( wp_get_nav_menu_items(get_nav_menu_locations()['menu-header'])) {
?>
<!-- Output some HTML -->
<?php
}
Code References:
wp_get_nav_menu_items( int|string|WP_Term $menu, array $args = array() )