How to control display of page lists on sidebar that doesn’t have childrens

Question

I would try replacing your $children definition, using get_page_children() or get_children():

<?php
global $post;
$children = get_children( array(
    'post_type'   => 'page',
    'post_parent' => $post->ID,
    'post_status' => 'publish'
) );
if ( $children ) {
    // Code to list child pages goes here
}
?>

Also, wp_list_pages() will always return a string (either populated or empty), so if you’re going to test against it, use:

if ( '' != $children )

…instead of just:

if ( $children )

Because the latter will always return true, since the return is an (empty) string.