Conditional statement for dates

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.';
    }
    ?>