instead of $currentdate = current_time('Ymd');
you can use $currentdate = new DateTime();
so the Final Solution should be as like given below
<?php
$currentdate = new DateTime();
$date = get_field('course_start_date', false, false);
$date = new DateTime($date);
if ($currentdate > $date) {
echo ' - Started.';
} else {
echo ' - Not started.';
}
?>