hide the author information from the non-member
If you copy the node.tpl.php file from the templates directory and create a node--mycontenttype.tpl.php file you should see the following around line 91:

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


Simply change it to this:

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


The $logged_in variable is a node template variable that has the value of TRUE if a user is logged in.

In this snippet you're just saying if the user is logged in, then display the submitted information.

Don't forget to clear your caches!

EDIT

Apologies, I've read the question properly now.  The following should do what you want.

&lt;?php if ($display_submitted): ?&gt;
  &lt;div class="meta submitted"&gt;
    &lt;?php print $user_picture; ?&gt;
    &lt;?php if ($logged_in == TRUE): ?&gt;
      &lt;?php print $submitted; ?&gt;
    &lt;?php else: ?&gt;
      &lt;?php print $date; ?&gt;
    &lt;?php endif; ?&gt;
  &lt;/div&gt;
&lt;?php endif; ?&gt;


This will display the entire submitted by text for authenticated users, and just the date for anonymous users.