Reinstalling from scratch didn't work for me. At first I thought the appliance maybe hadn't upgraded properly so I installed a new instance of VDP and re-created the backup schedules so there was nothing brought across from the upgrade appliance.
So far, rebooting every 48 hours is working for me so I wonder if there is some kind of resource leak. Annoyingly I now get alert emails telling me the services are stopped because vCenter has requested a guest reboot!
If the problems re-occur our plan is to try the earlier VDP 6.0.2 appliance and see if that still has the same issue. I don't really want to go back to 5.8 (which worked OK) as I don't think backing up ESXi 6 hosts is in the compatibility matrix, upgrading to VDP 6.1 was the first step in our vSphere 5.5 to 6 upgrade, so it hasn't gone well so far!
If you're struggling to get through even the first backup window are you able to allocate any additional RAM to the VDP appliance?