Refer the below sample code for your reference.
To set current date to datepicker you need to set setDate property also for datetimepicker you need to set defaultDate property with the date value.
If you need to modify height and width of datepicker then you need to change in css for table-condensed for height width you also need to specify the font-size as its dependent to font-size so it will adjust for height and width properly.
HTML
<html xmlns="http://www.w3.org/1999/xhtml">
<head id="Head1" runat="server">
<title></title>
<script type="text/javascript" src="http://ajax.googleapis.com/ajax/libs/jquery/1.9.1/jquery.min.js"></script>
<script type="text/javascript" src="http://maxcdn.bootstrapcdn.com/bootstrap/3.3.2/js/bootstrap.min.js"></script>
<link rel="stylesheet" href="http://maxcdn.bootstrapcdn.com/bootstrap/3.3.2/css/bootstrap.min.css" />
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/bootstrap-datepicker/1.5.1/css/bootstrap-datepicker.css" />
<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/bootstrap-datepicker/1.5.1/js/bootstrap-datepicker.min.js"></script>
<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/moment.js/2.15.2/moment-with-locales.js"></script>
<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/bootstrap-datetimepicker/4.17.45/js/bootstrap-datetimepicker.min.js"></script>
<script type="text/css" src="https://cdnjs.cloudflare.com/ajax/libs/bootstrap-datetimepicker/4.17.43/css/bootstrap-datetimepicker.min.css"></script>
<script type="text/javascript">
$(function () {
var dateNow = new Date();
$("#datepicker").datepicker("setDate", dateNow);
$('#datetimepicker').datetimepicker({
defaultDate: dateNow
});
});
</script>
<style type="text/css">
.table-condensed
{
width: 200px;
height: 10px;
font-size: 10px;
}
</style>
</head>
<body>
<form id="form1" runat="server">
<asp:ScriptManager ID="ScriptManager1" runat="server">
</asp:ScriptManager>
<asp:UpdatePanel ID="UpdatePanel1" runat="server">
<ContentTemplate>
<div class="row">
<div class="col-md-4">
<div id='datepicker' class="input-group date datepicker col-md-6" data-date="" data-link-field="dtp_input2"
style="max-width: 220px">
<input id="input" class="form-control" size="16" type="text" value="">
<span class="input-group-addon"><span class="glyphicon glyphicon-calendar"></span>
</span>
</div>
</div>
<div class="col-md-4">
<div id='datetimepicker' class="input-group col-md-6" data-date="" data-link-field="dtp_input2"
style="max-width: 220px">
<input id="Text1" class="form-control" type="text" value="">
<span class="input-group-addon"><span class="glyphicon glyphicon-calendar"></span>
</span>
</div>
</div>
</div>
</ContentTemplate>
</asp:UpdatePanel>
</form>
</body>
</html>
Screenshot
