Using get_posts or WP_query with similar $args:
$args = array(
'post_type' => 'your_custom_post_type',
'author' => get_current_user_id(),
);
$wp_posts = get_posts($args);
if (count($wp_posts)) {
echo "Yes, the current user has 'your_custom_post_type' posts published!";
} else {
echo "No, the current user does not have 'your_custom_post_type' posts published.";
}