I'd add a data attribute to your dropdowns that will allow you to identify and access them;
<asp:DropDownList ID="DropDownList1" class="dropdown" data-type="ddl_chapter" AutoPostBack="true" ...
For your js code you have to remember there may be more than one dropdown so if you want to access their individual properties you will need some kind of loop;
<script type="text/javascript">
$.each($("[data-type='ddl_chapter']"), function (index, item) {
var ddl = $(item);
alert('Drop down ' + index + ' has value ' + ddl.val());
});
</script>
If you are wanting to target them as a whole using jquery selectors then do something like
$("[data-type='ddl_chapter']").css("border-color", "red");