"Encrypt backups" disabled for Time Machine on Mavericks; due to partition?
According to this page: http://support.apple.com/kb/PH11102


  If the Encrypt checkbox is available, you can select it to keep your
  backup disk secure. If the Encrypt checkbox is dimmed, your backup
  disk doesn’t support encryption.
  
  Encryption is available only for Time Capsules and for partitions or
  disks attached directly to your computer and partitioned with the GPT
  partition scheme. If encryption is not available for your disk, put
  the mouse pointer on the dimmed checkbox, and after a moment an
  explanation appears. If a disk needs to be reformatted or
  repartitioned to support encryption, Time Machine prompts you to do
  so.


So check and see what the mouse has to say about why it is dimmed out the encryption option. It could well be that you need to reformat the drive