Change author field's empty behaviour
Unfortunately you've no way to find everywhere this info will appears, as it will depends of the modules/templates that you're using.

So you need to identify where it is displayed, and check if user id is equal to 0 ( 0 = Anonymous user).

You can do this in hook_preprocess for example, or just add a condition in your template.
Like for example the default node.tpl.php is like this:

&lt;?php if ($display_submitted): ?&gt;
  &lt;div class="submitted"&gt;
    &lt;?php print $submitted; ?&gt;
  &lt;/div&gt;
&lt;?php endif; ?&gt;


You could add this condition to hide the infos for anonymous user:

&lt;?php if ($display_submitted &amp;&amp; $uid != 0): ?&gt;
  &lt;div class="submitted"&gt;
    &lt;?php print $submitted; ?&gt;
  &lt;/div&gt;
&lt;?php endif; ?&gt;