$args = array(
'post_status' => 'publish',
'post_parent' => $post_id
);
$children = get_posts( $args );
if( count( $children ) > 0 ) {
//has children
} else {
//does not have children
}
Docs: get_posts()
This should also allow you to have access to the data of the children, should you need it. If you need any additional parameters in $args
(a post type, for example), make sure to include those.