1.    You have supported Web browser1.  WebClient service is started  

WebClient service which the WebDav protocol enables Windows-based programs such as Windows Explorer to create, access and modify Internet-based files.

Important Note:  If you are running a server type Windows OS, for example, you use Window Server 2008, installDesktop Experience feature first to get  WebClient service installed.

1. 1. From the Start menu, point to Administrative Tools, and click Services.
   2. Scroll down, right-click WebClient, and click Properties.
   3. In the Startup type list box, click Automatic.
   4. Click Apply.
   5. In the Service status section, click Start.
   6. Click OK.

For a list of supported Web browsers and support level information, please visit: 
Plan browser support
From the page you can see it’s recommended to use 32-bit Internet Explorer 7,8,9.

Note: “Open with Explorer” may not work if you have 64-bit Internet Explorer and 64-bit Microsoft Office installed.

3. Add https://*.sharepoint.com to Local intranet site

To configure Web browser, open “Internet Options” menu, navigate to the “Security Tab”, and addhttps://*.sharepoint.com to Local intranet site.

Then restart the browser and try it again.

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> If you have tried the above methods, however “Open with Explorer” still does not work, try to make sure you have applied the latest updates on the system.

> In addition, you can also try to enable Basic authentication on the client computer, by setting the BasicAuthLevel registry entry of the following key:

HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Services\WebClient\Parameters

For detailed steps, visit the following KB page.

You cannot connect to a document library in Windows SharePoint Services 3.0 or Windows SharePoint Services 2.0 by using Windows shell commands or by using Explorer View
http://support.microsoft.com/kb/841215