Stairs dont record
Which watch do you have?
So it get some but not all. A stair for Garmin is 10 feet/3 meters. How many 3 meter stairs do you climb and how many do the watch log?
Yes, you should be swinging your arm, so that the watch knows that you are walking, and not - for example - riding an elevator.
Also check your ascent at the same time. Every 3 meters going up stairs you will get 1 floor.