If you want to modify the appearance or behaviour of the drop-down list, you're going to need a JavaScript plugin.
Try
Select2[
^] - it will add a search box to filter your drop-down list.
@Html.DropDownListFor(x => x.IdGateSize, (SelectList)ViewBag.ListTypicalGates, "Please choose a gate ", new { @class = "js-select2", @style = "min-width: 500px;" })
<link href="https://cdn.jsdelivr.net/npm/select2@4.0.13/dist/css/select2.min.css" rel="stylesheet" />
<script src="https://cdn.jsdelivr.net/npm/select2@4.0.13/dist/js/select2.min.js"></script>
<script>
$(function(){
$(".js-select2").select2();
});
</script>
There are lots of options to let you customise the appearance and behaviour.
Since you're using Bootstrap, you might also want to apply the Bootstrap theme:
GitHub - ttskch/select2-bootstrap4-theme: Select2 v4 theme for Bootstrap4[
^]