Use the last updated date rather than publish date on posts.

genesis-byline.php

<?php 
add_filter( 'genesis_post_info', 'custom_post_info' );

function custom_post_info( $post_info ) {
    $u_time = get_the_time( 'U' ); 
	$u_modified_time = get_the_modified_time( 'U' ); 

	if ( $u_modified_time >= $u_time + 86400 ) { 
		$post_date = get_the_modified_time( 'F jS, Y' );  
	}
	else { 
		$post_date = get_the_time( 'F jS, Y' ); 
	}
	$post_info =  __( 'By', 'genesis' ) . ' [post_author_posts_link] ' . __( 'on ', 'genesis' ) . $post_date .' [post_comments] [post_edit]';
	return $post_info;
}