Maybe I'm missing something but this should be as easy as setting the ID attribute:
var $clone = $("#template").clone();
var index = $("table.reference tr:not(.template)").length;
$clone.attr("id", "emlRow_" + index);
UPDATES
var count = $("table.reference tr").length;
$("#Add").click(function() {
count++;
var $clone = $("#secondaryEmails tbody tr:first").clone();
$clone.attr({
id: "emlRow_" + count,
name: "emlRow_" + count,
style: "" // remove "display:none"
});
$clone.find("input,select").each(function(){
$(this).attr({
id: $(this).attr("id") + count,
name: $(this).attr("name") + count
});
});
$("#secondaryEmails tbody").append($clone);
});
working example: http://jsfiddle.net/hunter/EwQUW/35/