I have a couple of WordPress sites that live behind HTTP basic authentication—they're family-related blogs that I don't want publicly accessible.
Everything works fine behind the authentication except for automatic updates (and other WordPress cron tasks). This bugged me a bit, but never enough to do anything about it until today. To fix the problem, you can either modify some code (which I generally don't like to do) or use a plug-in (easy and quick). I chose the plug-in.
If you'd prefer the customization solution, though, just follow these instructions. I haven't tested them myself, though, so I can't say for sure that they still work (they are five years old).
For the plug-in solution, install WP Cron HTTP Auth. Go to its Settings panel, and enter your HTTP Authentication credentials, and that's that—WordPress cron tasks, including automatic updates, will now work again. Hooray!