Hi micah,
Check this example. Now please take its reference and correct your code.
HTML
<script type="text/javascript">
function ShowButton(ele) {
document.getElementById('btnSave').style.display = ele.value.length > 0 ? "block" : "none";
}
</script>
<asp:TextBox runat="server" ID="txtName" onkeyup="ShowButton(this)" />
<asp:Button ID="btnSave" Text="Save" runat="server" Style="display: none;" />
Screenshot