According to that page:
http://wpcodesnippet.com/hide-browser-upgrade-warning-wordpress/
You can put this code into your functions.php to get rid of that notices:
// hide browser upgrade warning in wordpress
function wcs_disable_browser_upgrade_warning() {
remove_meta_box( 'dashboard_browser_nag', 'dashboard', 'normal' );
}
add_action( 'wp_dashboard_setup', 'wcs_disable_browser_upgrade_warning' );
I would tell your client the risks that might be comming with this - just to make sure that you won't be responsible if anything happens.
Hope the code works - I couldn't test for my browsers are updated ;)