I was not able to use the Explorer view in SharePoint Online. I kept getting an error that said:

"Your client does not support opening this list with Windows Explorer"

I followed the following instructions:

1) Close all browsers

2) Click on the start button

3) Click “Run…” (You may have to go into the All Programs->Accessories)

4) Type Services.msc, click enter

5) Scroll down to WebClient

6) Stop the WebClient and then restart the WebClient

7) After it is restarted, close out of Services.msc

8) Open the 32-bit Internet Explorer browser

9) Go to the Team Site

10) Click on a Library

11) Under the Library Tools, select the Library tab

12) Click Open with Explorer

If all the latest updates are applied, the WebClient service is started, and the issue persists, make sure that an entry for SharePoint is added to the trusted sites zone in Windows Internet Explorer. To do this, follow these steps:

1) In Internet Explorer, on the Tools menu, click Internet options.

2) Click the Security tab, click the trusted sites zone, and then click Sites. Verify that the following entries are listed in the Websites list:https://*.outlook.com, https://*.sharepoint.com, https://*.microsoftonline.com, and https://*.lync.com

3) If these entries are not present, add them to the trusted sites list of websites.

4) In Internet Explorer, on the Tools menu, click Internet options.

5) Click the Security tab, click the Local intranet zone, click Sites, and then click Advanced.

6) Verify that the following entries are listed in the Websites list: *.microsoftonline.com, *.sharepoint.com, *.outlook.com, and *.lync.com

I got these instructions from the Office 365 Community forum:

http://community.office365.com/en-us/f/146/p/51607/180979.aspx#180979