Most of these questions are specific to the house and I hesitate to make generalities. For example, a staircase in the middle of a house could stir up the ‘center’ stars, which are not actually stationed in the literal center of a house, but they travel through the whole house. Like a vortex, the geometric center of a house can be important. If the “center” stars are inherently positive, then the extra activity of occupants dashing up and down the stairs could stimulate that energy even further. In my second book, The Feng Shui Matrix, I have a whole chapter on the “hidden agenda” of each house, the center energy.
As purveyors of energy, it will matter if the staircase has risers and the degree in which it is curved as that can manipulate qi flow are readily as furnishings and other décor items in the house. There is nothing inherently good for bad about carpeted stairs, but as a practical matter they would be safer and occupants less likely to slip on the stairs.
A very common floor plan for two level apartments and condos has the main entrance aligned with a staircase and little space to place buffers, like a plant or fountain, to slow down the direct path of qi.
A staircase that begins on a lower floor, turned away from the front door should be a non-issue. These are some of the basics covered in my case study on The Nature of Qi.