The style you like seems to be cantilevered into the wall for stability - depending on your available options you can use solid wood - but the more stable option is to clad it ! Just remember that not everyone likes the way that it's seems very open - so may limit your buyers if you plan to sell in the future !

is it a curved staircase? What is the highest point of the staircase? maybe you can do something like this image:

hi struti I would say it's matter of choice. marble or even granite or rough stone will have its own charm and definitely long lasting but only when staircase passage is comfortable huge. if space is compact wooden stairs would be better as its could be playfull with spiral or even basics with moulds and finishes, giving staircase elevation a contemporary look thanks shradha shah beescheese design

Just because you want to use the space below the stairs, do not just add drawers or shelfs unless you reaalllyyy need it. I would suggest go for adding something green to your house. Its healthy by adding air purifying plants. These plants are nowadays available in most of the nurseries and moreover Green patch in the house gives a beautiful look to the house and is also good for eyes ( the green color). Below are few picture to motivate you for Green patch or Indoor garden.