NuGetからWindows API Code Pack - Shellをインストール
Windows API Code Pack-Shell

MainWindow.xaml.cs

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using System.Windows;
using System.Windows.Controls;
using System.Windows.Data;
using System.Windows.Documents;
using System.Windows.Input;
using System.Windows.Media;
using System.Windows.Media.Imaging;
using System.Windows.Navigation;
using System.Windows.Shapes;
using Microsoft.WindowsAPICodePack.Dialogs;

namespace OpenDirectoryDialogSample
{
    /// <summary>
    /// MainWindow.xaml の相互作用ロジック
    /// </summary>
    public partial class MainWindow : Window
    {
        public MainWindow()
        {
            InitializeComponent();
        }

        private void OpenDirectoryButton_Click(object sender, RoutedEventArgs e)
        {
            var dialog = new CommonOpenFileDialog();
            dialog.IsFolderPicker = true;
            CommonFileDialogResult result = dialog.ShowDialog();

            if(result == CommonFileDialogResult.Ok)
            {
                var directoryPath = dialog.FileName;
                DirectoryPathTextBlock.Text = directoryPath;
            }
        }
    }
}

参考
wpf - Open directory dialog - Stack Overflow