The approach I've used very successfully is to set up an embedded Browsermob proxy server (in Java code) and register a RequestInterceptor to intercept all incoming requests (that match the host / URL pattern in question).
When you have a request that would otherwise need Basic auth, add an Authorization HTTP header with the credentials required ('Basic ' + the Base64-encoded 'user:pass' string. So for 'foo:bar' you'd set the value Basic Zm9vOmJhcg==)
Start the server, set it as a web proxy for Selenium traffic, and when a request is made that requires authentication, the proxy will add the header, the browser will see it, verify the credentials, and not need to pop up the dialog.
You won't need to deal with the dialog at all.
Other benefits:
- It's a pure HTTP solution, it works the same across all browsers and operating systems.
- No need for any hard-to-automate add-ons and plugins, any manual intervention.
- No need for custom profiles, custom preferences etc.
- You control credentials in your test code, and don't store them elsewhere.