First of all, I see a really large post there. Looks to be supporting a beam which supports the floor joists that terminate at the stair opening. You're not removing that are you?

If you are going contemporary why not try a solution similar to this - if you have mixed widths of slats you can disguise the beam by incorporating them into the design

I agree with working with the supporting beam. Perhaps add another post in the large gap further up the staircase, then use safety glass or thick Perspex sheeting to handrail height between the posts. This will allow light through, but provide the safety aspect of a bannister rail.

I actually did do this in a home, due to it being a supporting wall, we were unable to completely open it...we cut between the openings in the studs and created staggered openings all the way up...it was much better than closed....In this case I would use slats..and as Dytecture mentioned, u may have a support to work around.

Thanks for all the comments!! My preference is to remove both posts but I'm still looking for a structural engineer to determine if that's safe. Can anybody give a recommendation on someone in the Bay Area who could do this? If I can remove the posts, then I'm leaning towards just adding handrails... What's a relatively easy DIY way to finish the exposed stairs? The part I find difficult is finishing off the area where the lip of each step juts out. I've already stripped and sanded off the paint and would rather not replace the steps for a squared off finish.

If you want the treads to cantilever like the ones in your photo, then you'll need to replace them or add a additional tread on top of the existing. Keep in mind (regarding second option and if the stairs were constructed properly ) this raises the riser height on the bottom step and lessens the riser height on the very top riser. Properly constructed stairs have risers heights that are all equal after floor finishes have been installed on both bottom and top landings. A common tread thickness is 3/4" - 1 1/4". From your photo, I can't determine if head clearance is a issue when descending the stairs.

What part of the Bay Area? I'm in Sonoma County and have many engineers and architects available. If you are not North Bay, use the find local pros button on the header and look for a local architect...get a referral to an engineer or two.