Take a look at the following reminders screenshot and you’ll see there’s no plus button for adding a reminder.
I’d been banging my head against the wall trying to fix this, but I found a fix.
First, go to Settings > iCloud and turn Reminders on.
Now, go back into the Reminders app, and the plus sign should appear.
After this, you can go back and turn the iCloud Reminders setting back off if you’re concerned about battery usage. The plus sign will remain.
Another potential solution is to tap the list icon in the top left corner, then search for a list. The results will come up empty, but then you can tap the edit button and create a new list. This will also fix your reminders app.