I use it daily since I got original iPad three ?? years ago.
Since migrating to Newsstand (NS) the auto download works, at best, once a week. However, I'm an early riser, so I typically look at the app around 4:30 AM. It's possible the auto download occurs at a set time each day and maybe I'm checking the app too early. Today, I noticed the auto download was there but that's because I checked it at 5:45 (I sleep late on the weekends!).
As to manual updates, they always work, but if not, try deleting the app, reboot the device and download a fresh copy.
Only issue I noted (since moving to NS) is the paper freezes for a few seconds, but last week's update seems to have resolved it.
The tech team is very responsive, it's easier to get a reply by calling them (800) 568-7625. You'll get a low level techie, but you can ask to speak to level 2 and they'll patch you through. They also respond to emails, but usually that takes a few days, or more.
Other than the lack of auto downloads (which in my case might be a result of checking too early) the app is very stable.
That being said, the iPhone app never downloads unless I fire it up, but I only read WSJ on iPad so this isn't an issue. It also seems that on iPhone you get the Now version, not the Today version,
I'd reinstall and if that doesn't resolve, I'd call them.