How do I insert a user picture's thumbnail on the "Submitted by" field?
Hope this will help you.

In Drupal 7 you can achieve this using template preprocess functions - THEMENAME_preprocess_node.

Steps


First add default user picture, go to sites/default/files and create a folder user_pictures and add default user picture by name default.png 
Now add default picture URL using admin/config/people/accounts.


Add this code in template.php



    function THEMENAME_preprocess_node(&$variables) {
      $node = $variables['node'];
      if ($vars['view_mode'] == 'teaser') { //check for teaser views
        //get author picture uri
        $author_picture_uri = db_query("
        SELECT uri FROM {file_managed} fm
        INNER JOIN {users} u ON u.picture = fm.fid 
        WHERE u.uid = :uid", array(':uid' => $node->uid))->fetchField();
        $author_picture_uri = !empty($author_picture_uri) ? $author_picture_uri : 'public://default_user.jpg';
        $usr_pic = theme_image_style(
          array(
        'style_name' => 'thumbnail',
        'path' => $author_picture_uri,
        'attributes' => array('class' => 'author-pic'),
        'width' => NULL,
        'height' => NULL
          )
        );
        if ($variables['submitted'] && isset($node->uid)) {
          $variables['submitted'] = t('Submitted by !username !userpicture on !datetime', array('!username' => $variables['name'], '!userpicture' => $usr_pic, '!datetime' => $variables['date']));
        }
      }
    }



Where public://user_pictures/default.png equivalent to sites/default/files/user_pictures/default.png