Drupal Less cache keeps generating cache file that does allow access to the apache server
I do three things. 


I put myself in the apache group.
Set my umask and the httpd process to 0022
chmod sites/default/files to 2775


The first step lets me write to the directory. The second step makes sure that files I create have group write permission, and that apache does the same.  The third step tells the OS to always use the group of the directory owner. 

You should then be able to drush cc all and have Apache create the LESS files. 

For your initial fix, you may have to fix up the perms recursively. Eg,

chown -R apache.apache docroot/sites/default/files/
find docroot/sites/default/files/ -type d -exec chmod 2775 {} \;
find docroot/sites/default/files/ -type f -exec chmod 664 {} \;


I would also suggest reading Securing file permissions and ownership.  The comments have a good duscussion about the setgid bit and potential downfalls.