I am facing a similar issue. My website's WordPress installation is in the /cms subfolder under website root.
However, the cache html file generated by W3-Total-Cache removes the /cms part from all page resources (css, js, etc.), which is causing 404 errors for these files and therefore pages are not displaying properly.
I am attaching the index.html_ file from my w3tc/pgcache folder. As you can see from it, all .js, .css files etc. are being included as http://www.vijaypadiyar.in/wp-content/<path-to-file> instead of http://www.vijaypadiyar.in/cms/wp-content/<path-to-file>. As a result, my home page is messed up (see attached image).
Link to index.html_ generated by w3tc: https://docs.google.com/open?id=0B3fAqJpOwcyxcXNqa2NjTW5sOHM
Image of home page when problem occurs: https://docs.google.com/open?id=0B3fAqJpOwcyxeTFxMlF0a2N4QkE
Surprisingly, this seems to be happening only for the home page and not for other pages on my site. Also, it does not happen if I am logged in and viewing the uncached version of my site (I have disabled caching for logged-in users). Nor am I seeing the issue (yet) after disabling the W3-Total-Cache plugin.
This issue has started only since the last few days. I am not sure if nickrag's issue is the same.