Hi asp777,
Check the below example.
For jQuery Ajax refer below article.
For validation refer below article.
Controller
public class HomeController : Controller
{
    // GET: Home
    public ActionResult Index()
    {
        return View();
    }
}
View
@{
    Layout = null;
}
<!DOCTYPE html>
<html>
<head>
    <meta name="viewport" content="width=device-width" />
    <title>Index</title>
    <script type="text/javascript" src="https://ajax.googleapis.com/ajax/libs/jquery/1.8.3/jquery.min.js"></script>
    <script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/jquery-cookie/1.4.1/jquery.cookie.min.js"></script>
    <script type="text/javascript">
        $(function () {
            $("#btnLogin").on("click", function () {
                var rememberMe = $('#chkRememberMe').prop('checked');
                if (rememberMe) {
                    var userId = $('#txtUserId').val();
                    var password = $('#txtPassword').val();
                    // Make Ajax call to controller and validate user inputs.
                    // If valid userid and password set cookies in Ajax Success function.
                    $.cookie('userId', userId, { expires: 1 });
                    $.cookie('password', password, { expires: 1 });
                    $.cookie('rememberMe', true, { expires: 1 });
                }
                else {
                    $.cookie('userId', null);
                    $.cookie('password', null);
                    $.cookie('rememberMe', null);
                }
            });
            var remember = $.cookie('rememberMe');
            if (remember == 'true') {
                var userid = $.cookie('userId');
                $('#txtUserId').attr("value", userid);
                var password = $.cookie('password');
                $('#txtPassword').attr("value", password);
            }
            $("#txtUserId").on("input", function () {
                var remember = $.cookie('rememberMe');
                if (remember == 'true') {
                    var userId = $.cookie('userId');
                    var uId = $("#txtUserId").val();
                    if (userId == uId) {
                        var password = $.cookie('password');
                        $('#txtPassword').val(password);
                    } else {
                        $('#txtPassword').val('');
                    }
                }
            });
        });
    </script>
</head>
<body>
    @using (Html.BeginForm())
    {
        <table>
            <tr>
                <td>User Id</td>
                <td>@Html.TextBox("userId", "", new { id = "txtUserId", required = "required" })</td>
            </tr>
            <tr>
                <td>Password</td>
                <td>@Html.TextBox("password", "", new { id = "txtPassword", type = "password", required = "required" })</td>
            </tr>
            <tr><td colspan="2" align="center"><input type="checkbox" id="chkRememberMe" /><label>Remember Me</label></td></tr>
            <tr><td colspan="2" align="center"><input type="submit" id="btnLogin" value="Login" /></td></tr>
        </table>
    }
</body>
</html>