I have same problem. Health App records steps but not # of stairs climbed. I have tried all suggested remedies(reset phone, check privacy settings, etc.). I am surprised and disappointed at how immature the Apple Health software and its integration with hardware is. I just have a higher expectation for Apple and they appear to be years behind the Fitbit application and hardware. Very clumsy.
So now I have a $600 Apple Watch with inferior health tracking(no steps, no sleep) when compared to a Fitbit and a Citizens ECO powered watch that has infinite battery life for the watch, 4 days for the Fitbit and much more mature health tracking. I have to decide if the superb notifications on the Apple Watch are worth it. They might be.