In Internet Explorer (I’ve tested in all v6 – v9) the border on the sub-menu does not at first appear when you hover over the text. The second time you hover it will display. It works fine in Firefox and Chrome – i.e. it always displays the border. I’ve put an example page up on my site.
IE (on hover):

FF/Chrome (on hover):

There’s nothing fancy here, some css, onmouseover/onmouseout Javascript to set style.display = block/none. I’ve followed some of the ideas in this answer to a similar question.
I’ve stripped it down to the minimum to try and find the problem, but still no luck.
The sub-menu ul element has display:none set on it. It seems that IE doesn’t bother drawing the border until it has display:block set, and doesn’t draw it initially when Javascript is used to display the element.
<html>
<head>
<title></title>
<style type="text/css">
ul, li {padding:0; margin:0; border:0;}
ul#hover_menu_list,
ul#hover_menu_list ul {list-style-type:none;}
ul#hover_menu_list li {float:left;position:relative;display:inline;}
ul#hover_menu_list li ul {border:1px solid #000;display:none;position:absolute;left:0px;top:20px;width:170px;}
ul#hover_menu_list li ul li {display:block; clear:left; float:left;width:140px;}
</style>
</head>
<body>
<ul id="hover_menu_list" onmouseout="document.getElementById('menu1').style.display='none';" onmouseover="document.getElementById('menu1').style.display='block';">
<li>
Menu
<ul id="menu1">
<li>Submenu1</li>
<li>Submenu2</li>
</ul>
</li>
</ul>
</body>
</html>
UPDATE: The problem was indeed a doctype issue. Adding in either a transitional or strict doctype fixes the problem. The linked page has been updated with the fix.
As mentioned in the comment, it looks like you are missing a doctype.
Oldie but goodie: