Modify your CSS like this:
.vertical_banner {
border: 1px solid #E9E3DD;
float: left;
height: 210px;
margin: 2px;
padding: 4px 2px 10px 10px;
text-align: left;
width: 117px;
position:relative;
}
#bottom_link{
position:absolute; /* added */
bottom:0; /* added */
left:0; /* added */
}
<div class="vertical_banner">
<div id="bottom_link">
<input type="submit" value="Continue">
</div>
</div>