I am running C1-7 pro pretty much trouble-free since version 7.0.1 (with sessions, not with catalogues).
I feel 7.1.2 was a major step in terms of stabilty... and in the only recently released 7.1.3 version P1 seem to have fixed some (minor) issues again (at least crash after quitting the software seemed to be fixed). Currently I am on OS X 10.8.3.
Make sure to make a clean uninstall
of older Capture One versions: https://captureintegration.com/uninstalling-capture-one-4-on-mac/
The last chapter in the blog is critical: delete all older proxies (*.cop and *.cof files) prior to installing the new C1 version.
I would also recommend not to open on older sessions (sessions containing settings of former C1 versions that is)... but if you need to you should first try a session with only a few captures in it.
Finally I would recommend to set the image preview size (preferences, third tab) to a lower size (1024 or so... depending on your display and workflow).