This answer uses the jQuery UI Datepicker, which is a separate include. There are other ways to do this without including jQuery UI.
First, you simply need to add the datepicker class to the textbox, in addition to form-control:
<div class="form-group input-group-sm">
@Html.LabelFor(model => model.DropOffDate)
@Html.TextBoxFor(model => model.DropOffDate, new { @class = "form-control datepicker", placeholder = "Enter Drop-off date here..." })
@Html.ValidationMessageFor(model => model.DropOffDate)
</div>
Then, to be sure the javascript is triggered after the textbox is rendered, you have to put the datepicker call in the jQuery ready function:
<script type="text/javascript">
$(function () { // will trigger when the document is ready
$('.datepicker').datepicker(); //Initialise any date pickers
});
</script>