Check if author or current user has posts published

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.";
}