Do you have an interior wall in the same place above and below?
Interior is important because no insulation. Cut a small rectangle upstairs about the size and location of a 1-gang old work box. Use a 1/2? auger drill and extension to drill through the hole at a slight angle through the bottom plate of the wall.
Directly below that spot downstairs, cut another hole for a 1-gang old work box. Fish a weighted string down the wall and catch it in the basement cutout. Use that to fish your wire.
If you can do that, worst case you put a wall outlet on each hole and run patch cables to and from.
ETA: to run through duct work, get ?Plenum? rated cable.
Yes, it's an interior wall for both locations. The area I want to run to is directly below the area I want to run from.
"We've done the impossible, and that makes us mighty."
Nathan Fillion, "Firefly"