I would not paint any part of it. You would probably be hurting the value of the home by creating a restoration project for the next owner. Who would no doubt be looking for a discount in the sale price to pay for it. While not easily found you can have those moldings reproduced to repair areas that are damaged. Consider some nice area rugs to brighten up the room.

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Thank you for your replies, Sina and Jeff! The home was built in 1925 in New Jersey. The baseboards may be original but the window trim is not.

Based on your photos it looks like you have a mish mash of some original woodwork and for whatever reason, newer, poorly scaled and installed trim. For example, in your second photo, the baseboard along the long wall to the right looks to be #2 common pine that's been stained. It looks like the door leading to the basement (which appears to have been altered in height and/or replaced at some point) was trimmed with simple 1x flat stock that has no profile whatsoever. Same goes for the window at the bottom of the stairs except it looks like someone straight cut a piece of Big Box colonial casing and nailed it on for the apron. Given all that, I think your idea is solid to paint the woodwork and leave portions of the staircase stained. You might even consider replacing the door and window trim with paint grade moldings that are appropriate for your house and are properly proportioned and installed.

The newell post, balusters, railings, and skirt boards of the staircase look like untouched originals and would look terrific kept stained. If you go that route, I would recommend you wipe on your polyurethane rather than applying it with a brush. Its easier, faster and produces a finish that looks sprayed. If your desire is a satin finish, make sure your base coats are gloss and only use satin for the final layer(s). The reason being is that gloss is not only a harder and more durable finish, but it remains clear as multiple coats are applied. A finish will get muddy looking through multiple coats of satin due to the additives which dull the sheen. You can learn more about wiping on a poly finish from the website The Wood Whisperer. That's where I learned to make my own wipe on poly (easy - regular oil based poly diluted with mineral spirits - and much cheaper than buying ready mixed wipe on in a can).

Back to your question. Since the baseboards in around your room appear to be mix of old and a poorly selected species for staining (pine) I'd paint them all. You won't be devaluing your home. Someone already did that when they hacked into the original trim. What you'll be doing is repairing someone else's attempts at "remuddling" and adding value. The first picture that posted by Sina would work. Depending upon their condition and how much repair they might need, I might also suggest you consider painting the stair risers (NOT treads) as well. That's a personal choice but it's a classic look and can lighten the look of the stairs.

As for the door to the basement, it looks like it's been significantly altered or the original was replaced with a door that was too small. My first choice would be to replace it with a better fitting door. If that's not in the budget, and it doesn't have nice grain or is in good shape, I'd paint. It's a judgement call without seeing it in person.

Here's another picture that might give you some inspiration. Your staircase architecture is a touch different but I think you'll get the idea.

I would get an old door and put nicer trim around your windows.