I'd like to give my Sharp T816 TU-R60HA to a friend who has no experience with PVRs. It currently has V6.7 installed

Is the V3.7 noticeably more stable than V6.7?

I know hat V3.7 lacks series record and a 30 second skip but would prefer to opt for stability.

Yes, definitely more stable and more responsive. Once it is changed you'll never be able to get V6.7 back, although probably V6.8 is not much different. The biggest problem that the older software has is that often recordings end up being 2-3 hours longer than they should have been, due to a bug triggering an intentional Freeview+ fallback feature. I'm not sure if you have automatic updates enabled on your Sharp, but for me it does help with stability - with updates enabled, at least the box will reboot at 3am every day bringing some degree of stability with it on V6.8 (only occasionally does the box tend to lock up requiring the plug to be pulled).