csc /r:C:\Windows\Microsoft.NET\assembly\GAC_MSIL\UIAutomationClient\v4.0_4.0.0.0__31bf3856ad364e35\UIAutomationClient.dll ^
/r:C:\Windows\Microsoft.NET\assembly\GAC_MSIL\UIAutomationTypes\v4.0_4.0.0.0__31bf3856ad364e35\UIAutomationTypes.dll ^
/r:C:\Windows\Microsoft.NET\assembly\GAC_MSIL\WindowsBase\v4.0_4.0.0.0__31bf3856ad364e35\WindowsBase.dll ^
Test.cs

using System;
using System.Text.RegularExpressions;
using System.Runtime.InteropServices;
using System.Windows.Automation;


public static class NativeMethods
{
    [DllImport("user32.dll", CharSet = CharSet.Auto)]
    public static extern IntPtr FindWindowEx(IntPtr hWndParent, IntPtr hWndChildAfter, string lpszClass, string lpszWindow);

    public delegate bool EnumWindowsDelegate(IntPtr hWnd, IntPtr lparam);

    [DllImport("user32.dll")]
    [return: MarshalAs(UnmanagedType.Bool)]
    public extern static bool EnumWindows(EnumWindowsDelegate lpEnumFunc,   IntPtr lparam);
}

public class Class1
{
    static AutomationElement FindCredentialWindow()
    {
        IntPtr hWnd = NativeMethods.FindWindowEx(IntPtr.Zero, IntPtr.Zero, "Credential Dialog Xaml Host", "Windows セキュリティ");

        if ( hWnd == IntPtr.Zero ) {
            return null;
        }

        return AutomationElement.FromHandle(hWnd);
    }

    static string FindTextBlockMatches(AutomationElement aeForm, Regex r)
    {
        var elems = FindElementsByClassName(aeForm, "TextBlock");
        foreach ( AutomationElement elem in elems ) {
            var elemInfo = elem.Current;
            if ( r.IsMatch(elemInfo.Name) ) {
                return elemInfo.Name;
            }
        }
        return null;
    }

    static void InputToCredentialWindow(AutomationElement aeForm, string userId, string password)
    {
        AutomationElement aeUserId   = FindElementById(aeForm, "EditField_0");
        AutomationElement aePassword = FindElementById(aeForm, "PasswordField_1");
        AutomationElement aeOkButton = FindElementById(aeForm, "OkButton");

        if ( aeUserId != null && aePassword != null && aeOkButton != null ) {
            var vpUserId   = (ValuePattern)aeUserId.GetCurrentPattern(ValuePattern.Pattern);
            var vpPassword = (ValuePattern)aePassword.GetCurrentPattern(ValuePattern.Pattern);
            vpUserId.SetValue(userId);
            vpPassword.SetValue(password);

//  OKも自動で押す場合はこれ
//            var ipOkButton = (InvokePattern)aeOkButton.GetCurrentPattern(InvokePattern.Pattern);
//            ipOkButton.Invoke();
        }
    }

    static AutomationElement FindElementById(AutomationElement rootElement, string automationId)
    {
        var cond = new PropertyCondition(AutomationElement.AutomationIdProperty, automationId);
        return rootElement.FindFirst(TreeScope.Descendants, cond);
    }

    static AutomationElementCollection FindElementsByClassName(AutomationElement rootElement, string className)
    {
        var cond = new PropertyCondition(AutomationElement.ClassNameProperty, className);
        return rootElement.FindAll(TreeScope.Descendants, cond);
    }

    [STAThread]
    static void Main()
    {
        AutomationElement aeForm = FindCredentialWindow();
        if ( aeForm != null ) {

            // 対象サイト(?)のメッセージに応じて適宜変更. 改行とか合わせるのが面倒なので正規表現で探すようにした
            string text = FindTextBlockMatches(aeForm, new Regex("^サーバー(.*)が"));

            if ( text != null ) {
                Console.WriteLine(text);

                // 下記は ID, PASSWORD に応じて適宜変更。機密管理に注意すること。
                //  やらかし例：ソースにパスワード埋め込んだままGitHubなどでうっかり公開したりとか
                InputToCredentialWindow(aeForm, "userid", "password");
            }
            else {
                Console.WriteLine("Not found Credential window with the description.");
            }
        }
        else {
            Console.WriteLine("Not found Credential window.");
        }
    }
}