After research in the Internet I found finally the solution, here's the answer and the credits belong to him.

I'm not sure why the WebPart couldn't or can't detect the ASP.NET controls in var variables maybe there's some about the if conditionals...

Here's the full validation code that I'm using in the WebPart that I develop a few days ago.

<script type="text/javascript">

function validateForm() {

    //Declare variables.
    var emailPattern = /^[a-zA-Z0-9._-]+@[a-zA-Z0-9.-]+\.[a-zA-Z]{2,4}$/;
    /*For some unknown reason, fields to validate (if they're declared on variables type "var") are NOT detected.*/

    //Validate fields (ASP.NET fields).
    if (document.getElementById('<%= txtSubject.ClientID %>').value.length == 0) {
        alert("Field ''Subject'' is required.");
        document.getElementById('<%= txtAsunto.ClientID %>').focus();
        return false;
    } else if (document.getElementById('<%= txtEmail.ClientID %>').value.length == 0) {
        alert("Field ''Email'' is required.");
        document.getElementById('<%= txtEmail.ClientID %>').focus();
        return false;
    }
    else {
        if (!emailPattern.test(document.getElementById('<%= txtEmail.ClientID %>').value)) {
            alert("Field ''Email'' is invalid.");
            document.getElementById('<%= txtEmail.ClientID %>').focus();
            return false;
        }
        else{                   
            //Your code...
        }
    }
}</script>

N.B: The following code was tested and it works in pure HTML, SharePoint 2010 and SharePoint 2013 (Online).

After research in the Internet I found finally the solution, here's the answer and the credits belong to him.

I'm not sure why the WebPart couldn't or can't detect the ASP.NET controls in var variables maybe there's some about the if conditionals...

Here's the full validation code that I'm using in the WebPart that I develop a few days ago.

<script type="text/javascript">

function validateForm() {

    //Declare variables.
    var emailPattern = /^[a-zA-Z0-9._-]+@[a-zA-Z0-9.-]+\.[a-zA-Z]{2,4}$/;
    /*For some unknown reason, fields to validate (if they're declared on variables type "var") are NOT detected.*/

    //Validate fields (ASP.NET fields).
    if (document.getElementById('<%= txtSubject.ClientID %>').value.length == 0) {
        alert("Field ''Subject'' is required.");
        document.getElementById('<%= txtAsunto.ClientID %>').focus();
        return false;
    } else if (document.getElementById('<%= txtEmail.ClientID %>').value.length == 0) {
        alert("Field ''Email'' is required.");
        document.getElementById('<%= txtEmail.ClientID %>').focus();
        return false;
    }
    else {
        if (!emailPattern.test(document.getElementById('<%= txtEmail.ClientID %>').value)) {
            alert("Field ''Email'' is invalid.");
            document.getElementById('<%= txtEmail.ClientID %>').focus();
            return false;
        }
        else{                   
            //Your code...
        }
    }
}</script>

N.B: The following code was tested and it works in pure HTML, SharePoint 2010 and SharePoint 2013 (Online).