I got to the Downloads page, selected 7.3.1, pressed the Install button. It changes to installing, but nothing happens after that. No error message, no login prompt (shouldn't be anyway, since I was already logged into my developer account).
Unable to install Xcode 7.3.1
The button via https://developer.apple.com/download/more/ should initiate a download, not an install.
If that link isn't working, use the Mac App Store instead.