Use an empty element sized for the content as the background, and position the content over the blurred element.
#dialog_base{
background:white;
background:rgba(255,255,255,0.8);
position: absolute;
top: 40%;
left: 50%;
z-index: 50;
margin-left: -200px;
height: 200px;
width: 400px;
filter:blur(4px);
-o-filter:blur(4px);
-ms-filter:blur(4px);
-moz-filter:blur(4px);
-webkit-filter:blur(4px);
}
#dialog_content{
background: transparent;
position: absolute;
top: 40%;
left: 50%;
margin-left -200px;
overflow: hidden;
z-index: 51;
}
The background element can be inside of the content element, but not the other way around.
<div id='dialog_base'></div>
<div id='dialog_content'>
Some Content
<!-- Alternatively with z-index: <div id='dialog_base'></div> -->
</div>
This is not easy if the content is not always consistently sized, but it works.