The Apple Watch does not count any stairs for me. I kept track for several days and go up and down apprximately 7-12 flights of stairs per day.

Hi

Firstly, check that you have set up the Activity app:

Track daily activity

If Flights Climbed are not being recorded as expected, then it may help to follow these steps:

If you have not yet done so, update your iPhone and Apple Watch to the latest software, starting with your iPhone:

• Update the iOS software on your iPhone, iPad, or iPod touch - Apple Support
• Update the software on your Apple Watch - Apple Support

Close certain apps and restart both your iPhone and your Apple Watch:

• Close the Watch app, Activity app and Health app (if open) on your iPhone:
  • Double-click the Home button, then swipe up on each app preview to close the apps.
• Turn both devices off together, then restart your iPhone first:
  • Restart your iPhone, iPad, or iPod touch - Apple Support
  • Restart your Apple Watch - Apple Support

Check that Wrist Detection is turned on and that, if passcode is enabled, you are unlocking your watch after placing it on your wrist

• On your iPhone, in the Watch app, go to: My Watch (tab) > General > Wrist Detection - turn on.

On your iPhone, go to: Settings > Privacy > Motion & Fitness - check that both Fitness Tracking and Health are enabled.

On your iPhone, in the Watch app, go to: My Watch > Privacy - check that both Fitness Tracking and Heart Rate are enabled.

Activity-related estimations depend, in part, on your personal information. To check that this is accurate:

• On your iPhone, in the Watch app, go to: My Watch > Health > Edit - tap and adjust items, then tap on Done.

Check that you are wearing your Apple Watch sufficiently snugly on the top of your wrist (if it is worn too loosely, your watch may believe that it has been removed from your wrist).

• Your heart rate. What it means, and where on Apple Watch you’ll find it. - Apple Support
• Wrist Detection can be affected by temporary or permanent skin changes, such as by some wrist tattoos.

Calibrating your watch improves the accuracy of various activity-related estimations. Instructions for how to calibrate your watch are available via the support article linked below. The article also includes instructions for resetting existing calibration data and starting afresh. (Resetting your calibration data will not erase your Activity history).

More information:

Use the Activity app on your Apple Watch - Apple Support

Calibrating your Apple Watch for improved Workout and Activity accuracy - Apple Support