What Java version do I need for Minecraft?

Simply download the latest version that's compatible with your system from the official Java website.
The big red 'Download' button will automatically forward you to the right file.

The official Minecraft system requirements list Java 8 as the minimum version to run Java.

But for a few years already, the launcher downloads Java for you if you don't have it already, meaning that a missing Java installation would be a launcher bug. But before you report it, please make sure that it's actually that and not a crash or other problem.

The launcher is automatically downloading java 8. If you are getting crash report please post it here. If the crash report is "Failed to write core dump. Minidumps are not enabled by default on client versions of Windows" update your videocard and if the error is still showing, i can't help you. If Minecraft 1.0-1.12 is crashing it's because of fbo. Make sure to disable it from options.txt in the .minecraft folder.