For me, it depends. In 3.7 while the feature is still in its infancy, I plan on rolling auto-updates on everything on my personal blog and sites that I manage for other people. I've been beta testing auto-updates since the feature was first pushed into core on my own blog (yeah, I live on the edge). I haven't had a single issue.

On business sites that I manage, I'll disable this feature. I want to make sure everything is in perfect order before updating and wouldn't want to run the risk, however minimal, of something going wrong when I'm not around to fix it.

Automatic updates will make things much easier for me though. I manage many WordPress installs for friends and family. Some of them already handle their own updates with the "one-click" updater, but this will be even better. I pretty much won't have to do anything for them at all except for making sure I don't break the server.

I think some of those of us who are or have been hesitant about auto-updates were around back in the day when WordPress updates actually did break sites. Several years ago, WordPress completely messed up my database on an update. Of course, I had a backup and was able to restore from it. For anyone who has ever had that or something similar happen, the idea of automatic updates is a bit terrifying.