Comments pagination on author page

Make sure you’ve set the paged parameter in your query, below is shown how to do that.

<?php
$paged = ( get_query_var( 'paged' ) ) ? get_query_var( 'paged' ) : 1;
$args = array(
     'posts_per_page' => 3,
     'paged' => $paged
); 
$the_query = new WP_Query( $args ); 
?>

For more information about pagination have a look at this page.