There is a less 'hacky' solution than the accepted answer available here: Minimize a folder

It's based on the Shell Objects for Scripting. Sample:

const int SW_SHOWMINNOACTIVE = 7;

[DllImport("user32.dll")]
static extern bool ShowWindow(IntPtr hWnd, int nCmdShow);

static void MinimizeWindow(IntPtr handle)
{
    ShowWindow(handle, SW_SHOWMINNOACTIVE);
}

//call it like:

foreach (IWebBrowser2 window in new Shell().Windows())
{
    if (window.Name == "Windows Explorer")
        MinimizeWindow((IntPtr)window.HWND);
}

The same thing can be achieved using the Internet Explorer Object Model

// add a reference to "Microsoft Internet Controls" COM component
// also add a 'using SHDocVw;'
foreach (IWebBrowser2 window in new ShellWindows())
{
    if (window.Name == "Windows Explorer")
        MinimizeWindow((IntPtr)window.HWND);
}