My chop saw has a place to attach a shop vac.

Most miter saws have a provision for attaching a shop vac but it's far from perfect. Festool's Kapex is supposed to have the best dust collection of any saw but it's very, very expensive. There are also any number of aftermarket "shrouds" or "hoods" for miter saws too. The good news is it's only saw dust and pretty easily vacuumed off the floor.

You can only avoid sawdust in the room by doing the cutting somewhere else ...

To minimize the number of trips up and down the stairs, mark as many sections as you can, haul them down, cut them and bring them back up. Attach them and mark the next set of cuts.

Use a hand saw and miter box. You will still have sawdust, but it will be large grains and localized to the box area. A shop vac will be all that is needed to clean up.

Put an old sheet or a clean dropcloth under the miter box and you will be able to contain most of the sawdust and roll/fold it to remove it from the house.

or, hire someone else and let them cut in the garage and climb the stairs.

The regulars here have good ideas. Another is to just use a hand miter box. That would be faster than running up and down 3 flights of stairs.

Plug your saw into a Festool vacuum and connect the vacuum hose to the saw. The vac starts every time you pull the saw trigger. Virtually no dust.

Festool? Really? $700 for a shop vac for an occasional or one-time use? And it won't work any better than the weakest link, that would be the geometry of the miter saw. Sure, if I was a full-time trim carpenter doing rehab work every day, I'd spring for it. But not for casual use. #drinkingthegreeenkoolaid

Make a cut list and cut outside/garage. One trip.

Casey

I ran baseboard for years as Casey described... measure your cuts, make a list, go to remote cutting area, go back and nail.
We also used to use this same method on a two man production team, one guy "nailer" makes list for room or next few sections, hands list to "cutter", cutter brings back and scatters ahead of nailer.. repeat/repeat/repeat to do a complete house.
Same method works equally well for casing and crown molding.
On new construction as well as major remodels we would typically have a "shop" set up in a garage or large room for all cutting and sawdust, this way painters as well as other trades can continue working...