If you see the DOM element generated by bootstrap-select, it actually hides the original select element and creates its own structure with div,span and buttons. There isn't any official documentation provided to change its border, but as a workaround you could do it as below:
You need to change border of the button whose class is btn dropdown-toggle btn-default, which is generated by bootstrap-select. Since the element created will not be given any id we will make use of its class with document.getElementsByClassName("classnames")[0]
document.getElementsByClassName("btn dropdown-toggle btn-default")[0].style.borderColor = "red";
Update
Explanation in comments.
$(document).ready(function() {
//add a change event to each selectpicker
$('.selectpicker').selectpicker().on('change', function() {
var id = $(this).attr('id'); //get current element's id
var selected = $(this).find("option:selected").val(); //get current element's selected value
var condition = false; //default false to form valid
$(this).selectpicker('setStyle', 'btn-danger', selected == "" ? "add" : "remove");
if (id == "box") {//if its first select box
condition = !(selected == 2 || selected == ""); //check if its value==2 or value=="" so that form validation depends of respective selects
$("#box2").selectpicker(selected == 2 ? "show" : "hide").val("");
//show or hide box2 based on selected val
$("#box2").selectpicker('setStyle', 'btn-danger',"remove");
//remove danger from 2nd selectpicker when its hidden or shown
} else {
condition = !(selected == "");
//if its box 2 check if any value is selected
}
$(this).closest('form').data('valid', condition);
//set it to form's data-valid attribute
});
$("#box2").selectpicker('hide');//default hide on page load
//form submit event instead of validate() inline function
$("form").on("submit", function(e) {
e.preventDefault();//prevent default action of submit
var isValid = $(this).data('valid');//check if its valid
if (!isValid) {
//if not valid loop through each selectpicker
$.each($('.selectpicker'),function(){
var selected=$(this).val();
//check appropriate values for each select and set/remove the btn-danger class for respective selectpickers
$(this).selectpicker('setStyle', 'btn-danger', selected == "" ? "add" : "remove");
})
}
else{
alert(isValid);
//Submit your form
}
})
});
<link href="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/css/bootstrap.min.css" rel="stylesheet" />
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/bootstrap-select/1.10.0/css/bootstrap-select.min.css" />
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/js/bootstrap.min.js"></script>
<!-- Latest compiled and minified JavaScript -->
<script src="https://cdnjs.cloudflare.com/ajax/libs/bootstrap-select/1.10.0/js/bootstrap-select.min.js"></script>
<!--add data-valid attribute to form to make sure its valid or not and default to false-->
<form action="#" data-valid="false" method="POST">
<select id="box" name="num" class="selectpicker form-control">
<option value="" selected="selected">select number</option>
<option value="1">1</option>
<option value="2">2</option>
</select>
<select id="box2" name="num" class="selectpicker form-control">
<option value="" selected="selected">select number</option>
<option value="1">1</option>
<option value="2">2</option>
</select>
<div class="text-center">
<button type="submit" class="btn btn-primary">submit</button>
</div>
</form>
