Please refer this.
<html xmlns="http://www.w3.org/1999/xhtml">
<head id="Head1" runat="server">
<title></title>
<script type="text/javascript">
function SelectOrEnterText(src, args) {
var atLeast = 1;
var text1 = document.getElementById("<%=TextBox1.ClientID %>").value;
var text2 = document.getElementById("<%=TextBox2.ClientID %>").value;
var CHK = document.getElementById("<%=CheckBoxList1.ClientID%>");
var checkbox = CHK.getElementsByTagName("input");
var counter = 0;
for (var i = 0; i < checkbox.length; i++) {
if (checkbox[i].checked) {
counter++;
}
}
if (text1 == '' && text2 == '' && atLeast > counter) {
args.IsValid = false;
}
}</script>
</head>
<body>
<form id="form1" runat="server">
<div>
<asp:TextBox ID="TextBox1" runat="server"></asp:TextBox>
<br />
<asp:TextBox ID="TextBox2" runat="server"></asp:TextBox>
<br />
<asp:CheckBoxList ID="CheckBoxList1" runat="server">
<asp:ListItem Text="Mango" />
<asp:ListItem Text="Banana" />
<asp:ListItem Text="Apple" />
</asp:CheckBoxList>
<asp:CustomValidator ID="CustomValidator1" runat="server" ClientValidationFunction="SelectOrEnterText"
Display="Dynamic" ErrorMessage="Please Select Item From RadioButtonList or Enter Text in any one of the TextBox">
</asp:CustomValidator>
<br />
<asp:Button ID="Button1" runat="server" Text="Validate" />
</div>
</form>
</body>
</html>
Thank You.