Interestingly, my wife has an iPhone 5s (iOS7), a 2006 MacBook Pro which can't run anything newer than Mac OS 10.6.8, and she is running iTunes 10.7. When she syncs her phone via USB, she starts iTunes, select-checks Backup To This Computer, then connects the iPhone and syncs. After the backup and sync is finished, she disconnects the phone, then selects-checks Backup To iCloud. She did purchase extra storage space on iCloud so she could sync and backup to iCloud.
She gave me a new iPhone 5s for Christmas. I haven't activated it yet. When I do, I'll try doing it the way she does and report back on my findings. My computer is a mid-2010 iMac running 10.8.4.
I've been holding off upgrading my iPad 2 to iOS 7 because of the supposed hard requirement that iOS 7 requires iTunes 11.1 or newer. Perhaps it's not an absolute. I was also holding off upgrading my iPhone 4 to iOS 7 because of all the negative comments in another thread I have about iOS 7 running slow on an iPhone 4. With the new iPhone 5s, my adventure is about to begin trying to run iTunes 10.7 with it.