How do I treat a recently sanded pine staircase?
I have remodeled quite a few Victorians the ones with pine or fur treads were usually stained with a honey or amber colored stain then polyurethane, I used varathane a solvent based poly, I have been told the water based polys are as good but my limited experience with water based was not as good as solvent based. 

I use very expensive brushes this is the real trick to a nice finish with poly ($20.00 is normal for a good quality brush)  to make the brush last years you will need a container that you can put some solvent in with the brush and seal it, after cleaning most the poly out, put the brush in solvent at least 3/4 of the bristles covered ) 
Seal the jar next time you want to use a day week or year later take the brush out, shake out the excess solvent seal the jar up and start applying , when you mix your poly stir never shake (unless you have a vacuum top to pull the air bubbles out).
When done clean the brush add some fresh solvent to the jar and put your brush in. Do this if even just going for a lunch break once the poly drys the brush is toast.

Poly works best with a medium heavy coat and let dry completely, light sanding between coats for awesome wood grain. First I would stain the entire staircase then 1 coat on the entire staircase. Since you want a carpet runner I would then put down 1-2 more coats on each end an inch or 2 inside where the carpet runner will go. This really brings that old growth wood grain out.