this.radPdfViewerNavigator1.SaveButton.MouseDown += SaveButton_MouseDown;

private void SaveButton_MouseDown(object sender, MouseEventArgs e)
{
    SaveFileDialog savefile = new SaveFileDialog();
    savefile.FileName = "MyFileName";
    savefile.Filter = "PDF files (*.pdf)|*.pdf|All files (*.*)|*.*";

    if (savefile.ShowDialog() == DialogResult.OK)
    {
        this.radPdfViewer1.PdfViewerElement.SaveDocument(savefile.FileName);
    }
}