There is a login box on the Wordpress admin area login page to make sure this is always checked use the following snippet. Only use this if you know the only time admin users access the admin area is going to be on a private computer. Example of using it is http://www.paulund.co.uk/set-remember-me-to-be-always-checked

wordpress-always-checked-remember-me.php

<?php

function login_checked_remember_me() {
    add_filter( 'login_footer', 'rememberme_checked' );
}
add_action( 'init', 'login_checked_remember_me' );

function rememberme_checked() {
	echo "document.getElementById('rememberme').checked = true;";
}
?>