well, yea, naturally running stairs will increase your stamina, but there will come a stage where you can't get ur HR much lower. Also because running stairs will often be at your anaerobic limit your muscles will experience a big increase in acid build up.
Put it this way, after a weight session running stairs is one of the worst things you can do, because it will prolong your recovery time. So for starters after a workout you should do cardio in an aerobic zone (you can run up the stairs at a slower pace, as long as you don't reach your anaerobic barrier). Running in an aerobic zone helps you muscles remove the acid build up, thus shortening your recovery period. It's what is known as an active recovery.
Doing a proper stair workout as you discussed on pure cardio days is great, and I actually think it is good. A lot of people, in fitness centres especially, tend to stick to the low aerobic zones, and I think if you are young enough you should push the envelope and train in a high zone a lot of the time.
But as mentioned, if you want to make proper progress you will also need at least one pro-longed run a week where you run at a lower HR, ideally for around 40 mins. This increases your individual muscle cell's mitrochondride numbers (sorry about the terms, I studied all this in German...), which in term enables your body to convert more glycogen into energy. This of course is also of huge benefit when you run at a higher HR, as you can stay in an aerobic zone for longer. As a result you can lower your HR a lot more than you could by just doing stair workouts (or any anaerobic workout for that matter). Most make the mistake thinking this kind of topic should only be of interst for marathon runners, but that's not the case. My running goal for instance is to break the 3 mile 18 min limit, but in order to do that I know that I also need to keep up my 40 min runs. Without them I'd still be trying to crack the 19 min barrier.
It is something you have to experience really to believe it, but combining both makes a huge difference while running. If you have ever gone from a sitting job to an active job (tending bar for example, in a smoke free environment) you will have experienced a similar effect on your runs.
As mentioned though, I guess most people here are interested in building muscles. So after weight sessions you should definitely not run at or above your anaerobic barrier. But doing any aerobic activity within your limits will be of great benefit and aid your recovery after workouts!
