с value все правильно.
Мой код:
<div class="param-value">
<asp:Label ID="txtEmail" runat="server"></asp:Label>
<input id="Button1" type="button" value="button" onclick="splitEmails()" />
<input id="txt" type="text" value="hello" />
</div>
<script>
function splitEmails() {
var strEmails = document.getElementById('<%=txtEmail.ClientID %>').value;
var txt = document.getElementById('txt');
txt.value = strEmails;
}
</script>