I'm getting the same problem. And it all started with the latest plugin update on 5/22/2013. Got a message from Google saying:
"Google detected a significant increase in the number of URLs we were blocked from crawling due to authorization permission errors."
Tried to fetch our home page using Google Webmasters Fetch tool:
This was the report:
HTTP/1.1 403 Forbidden
Date: Tue, 28 May 2013 17:43:36 GMT
Server: Apache/2.2.20 (Unix) mod_ssl/2.2.20 OpenSSL/0.9.8m DAV/2 mod_auth_passthrough/2.1 mod_bwlimited/1.4 FrontPage/184.108.40.20635
Expires: Tue, 28 May 2013 18:43:36 GMT
Cache-Control: max-age=3600, public, must-revalidate, proxy-revalidate
X-Powered-By: W3 Total Cache/0.9.2.11
Last-Modified: Thu, 01 Jan 1970 00:00:00 GMT
Keep-Alive: timeout=5, max=100
Turned off W3TC completely, and the problem went away. Googlebot fetched the page with no problem.