I was consistently getting an error that rewrites were not enabled on my server using WordPress 3.7.1 and the latest version of W3TC, however they were in fact working correctly. After reviewing the code I noticed that the check in MinifyAdminEnvironment.php for verify_rewrite_working() and test_rewrite() require a rewrite rule that is missing as it is included in .htaccess for Apache - this is not supported for Nginx. Adding the following rule to my Nginx configuration cleared up the issue:
rewrite /wp-content/cache/minify/[^/]*/w3tc_rewrite_test$ /wp-content/plugins/w3-total-cache/pub/minify.php?w3tc_rewrite_test=1 permanent;