Try hanging some drapes or blankets over the door(s) at the top and bottom of the steps, to be sure that's where the sound is getting through. It may be through the floors.

Thanks. there are no doors at the top and bottom. It's a dramatic open staircase. I'm really hoping for a tech solution that doesn't involve redecorating -- like some kind of white/pink noise generator. But I don't know if they even work.

Would it be feasible to frame in a wall somewhere at the bottom and put in glass doors and panels?

Hanging drapes or blankets? Certainly not at the top, that would be dangerous. Such an approach isn't going to be featured in Interior Design magazine.

There is no way to frame in a wall without destroying the beautiful open aesthetic. I've been in an office full of short walled cubicles where they used pink noise, and you really couldn't hear the people in the next cubicle. I was wondering if this technology exists in a smaller, cheaper way to use in our home.

Open floor plans and noise control work at cross purposes. You can have one, or the other, but often not both at the same time.

It sounds like a design layout where function was sacrificed for appearance. So, you're stuck.

I have not heard of using pink noise but that doesn't mean anything. There are noise cancelling headphones that have a microphone on the outside to detect ambient noise, which is then 'inverted' by the circuitry and fed into the headphones at the appropriate volume to exactly cancel the incoming noise. I don't know if this technology might be available to drive speakers rather than headphones. If so, you could install an active noise cancelling system in the stairwell. With the mike at the downstairs end, and speakers pointing up the stairs to cancel out the sound coming from the basement.

Search for 'noise cancellation systems'. The pink (or white) noise generators are more of a masking, like using a perfume to cover odors. Cancellation is what I described above. You'll find a Wiki entry on it and also some vendors and products. This stuff can get pricey but keep digging and you might find something reasonable.