You need to add an <option>
before you begin your loop.
function query(){
// This is the line you are missing
echo '<option>Select a class</option>';
$classes = mysql_query("SELECT * FROM class") or die("Could not search!");
while ($row = mysql_fetch_array($classes)) {
echo '<option value="Select Class"'>' . $row['class_name'] . '</option>';
}
}