You need a server side code in order to use Session. The following code doesn't requires code behind file, but again the code inside script will run at server side.
<asp:DropDownList ID="ddl" runat="Server"
DataSourceID="sqlDS"
DataValueField="ID"
DataTextField="TEXT"
AppendDataBoundItems="true"
OnSelectedIndexChanged="ddl_SelectedIndexChanged"
AutoPostBack="True">
<asp:ListItem Text="" Value="" Selected="True" />
</asp:DropDownList>
<asp:SqlDataSource ID="sqlDS" runat="Server"
SelectCommand="spDS" SelectCommandType="StoredProcedure" />
<script runat="server">
protected void ddl_SelectedIndexChanged(object sender, EventArgs e)
{
Session["SelecteValue"] = ddl.SelectedValue;
}
</script>
Note: Make sure AutoPostBack="True"
for DropDownList.