Try 

    <?php 
    if ( !empty( array ( $curauth->facebook ) ) || 
         !empty ( array ( $curauth->linkedin ) ) || 
         !empty( array( $curauth->twitter ) ) )
    { 
        echo 'echo me if any $curauth info exists'; 
    } 
    ?>

<sup><b>Note:</b> This can be all on fewer lines, I've just put in additional line-breaks to make it all fit to avoid a horizontal scrollbar.</sup>

<hr>

__Update__:<br>
Reading up on [Author Templates][1] made me realise that once you've set the `$curauth` variable, e.g.

    $curauth = (get_query_var('author_name')) ? get_user_by('slug', get_query_var('author_name')) : get_userdata(get_query_var('author'));

you should be able to use this instead:

    <?php 
    if ( !empty ( $curauth->facebook ) || 
         !empty ( $curauth->linkedin ) || 
         !empty ( $curauth->twitter ) )
    { 
        echo 'echo me if any $curauth info exists'; 
    } 
    ?>


  [1]: http://codex.wordpress.org/Author_Templates