The first way to improve lower body strength in stair-workouts is to climb every other step (essentially performing a lunge). Because doing this puts your legs in a mechanical disadvantage your muscles have to get stronger to overcome this disadvantage. I also have certain people climb every three steps to challenge them even further. But this is usually left to very tall people or last people with very long legs. Two steps is usually enough to challenge most anyone.
For the purpose of strength improvement it's only necessary to do 1 to 5 sets of 1 to 30 repetitions on each leg. Yon can either alternate or use 1 leg at a time. As your strength improves you can either increase repetitions, resistance (via dumbbells, barbells, weighted vests, sandbags etc?), or speed to challenge yourself further.
Although I've done 20 flights of stair lunges totaling around 60 repetitions each leg it's important to remember the goal of the workout (are you trying to improve strength or strength endurance). Strength protocols usually only call for between 1 & 20 repetitions for a few sets. But feel free to experiment as nothing is written in stone.
One day I decided to see how hard it would be to climb every other step to the 20th floor. What a workout that was! Is it really strength training in technical terms?
No, but if you're able to do this you will have achieved a high level of strength endurance that many people will never achieve.
Another way to do stair lunges is to use diagonal, cross-over and side lunges. The great thing about using these lunges is they address exercising in different planes of motion. Most people exercise in straight line forward or what experts call the sagital plane. Most people have plenty of strength in this plane. It's in the side to side (Frontal plane) and the rotational (Transverse plane) that people are weakest in. It's no coincidence that people get hurt in these planes. Most ankle injuries occur in the frontal plane and many knee injuries occur in the transverse plane.
Stair climbing lends itself very well to doing lunges in all planes of motion and can really help someone improve their strength in all planes of motion to balance out their strengths and weaknesses. In my own experience I've went for weeks with just doing stair-climbing in frontal and transverse planes and my knees have never felt better.
Walking down stairs can also be used to improve eccentric and deceleration strength which is very important in everyday life as well as sports. Always remember to use caution when going down stairs (because of the risk of falling). I prefer to just use the first couple of steps and just work down from there. For obvious reasons one should never run down the stairs (it's too dangerous). Always remember safety first.
A lower body strength workout can look something like this:
Note: in this workout I had access to 20 flights.
Round 1: warm-up: Walk up 20 flights. Stretch when you reach the top. Take elevator down.
Round 2: Do 30 stair-lunges: 15 on weaker side followed by 15 on your stronger side. (Climb 2 steps each lunge).
Round 3: Do 30 stair-lunges in alternating fashion. (Climb 2 steps each lunge).
Round 4: Do 30 crossover-lunges in alternating fashion (climb 2 steps each lunge).Note: some people may not have the strength and/or flexibility to climb 2 steps in this fashion. In this instance just climb 1 step each lunge.
Round 5: Do 30 crossover lunges from side: 15 on weaker side followed by 15 on your stronger side. (Climb 2 steps for each lunge).Note: some people may not have the strength and/or flexibility to climb 2 steps in this fashion. In this instance just climb 1 step each lunge.
Round 6: Do 30 side to side lunges in alternating fashion. (Climb 2 steps for each lunge).Note: some people may not have the strength and/or flexibility to climb 2 steps in this fashion. In this instance just climb 1 step each lunge.