How can I prevent posts with no title showing up in my query?

Please find below a quick solution to your problem

1. first query for all empty titles
2. place the result in an array

3. query again for post__not_in
   see below sample code

   $query = new WP_Query('post_title=\'\'');
   $a_empty_titles = array();
    while($query->have_posts()){
        $query->the_post();
        array_push( $a_empty_titles ,$query->post->ID);
    }
    wp_reset_postdata();
    wp_reset_query();
    echo count($a_empty_titles);
    $query = new WP_Query( array( 'post__not_in' => $a_empty_titles ) );
    //your loop code here
    //reset query
    wp_reset_postdata();
    wp_reset_query();
    ?>