This will work....
<asp:dropdownlist cssclass="ddl" id="ddlGraduateYear" runat="server" xmlns:asp="#unknown" />
CSS..
.ddl {
width: 70%;
border: 2px solid #c7c7c7;
height: 35px;
padding-left: 0px;
padding-right: 0px;
padding-bottom: 0px;
padding-top: 0px;
border-radius: 0px;
display: inline-block;
-webkit-appearance:none;
background-image: url('../Images/CommonImages/dropdownarrow.png');
background-position: right;
background-size: 25px 25px;
background-repeat: no-repeat;
text-indent: 0.01px;
text-overflow: '';
vertical-align:middle;
}