There's a technology called X10 where you can control a light remotely by bypassing the switch in the wiring layout (wire the incoming and outgoing wires together so the switch controls nothing) and then using a remote to operate the light. It's an older technology based on radio waves. There are also some new wireless smart controls I don't know much about, but those are mostly for lamps. The X10 can work with a special receiver that screws in where the ceiling light bulb goes. People still use it for the very purpose you are needing here.

Could you submit a sketch? Is the door at the bottom of the stairs? Hinged on left or right? Are you able to put in a new switch box (or 2)?

SIlverlined: Thx. I'll look into that.

mtvhike: yes, the door is at the bottom of the stairs; hinged on the left with the switch on the same side as the hinges. I have to walk around the open door (in the dark) to reach the switch. No place to put the switch to the right of the door and the walls/ceilings are already finished.

1. Cut a hole in the staircase wall on the left-hand side for an old-work electrical box. Be sure there are no studs between this hole and the electrical box that houses the existing basement light switch.
2. Remove a knockout in the existing electrical box on the side that faces the hole you cut in step 1.
3. Remove the old single-pole switch that operated the lights.
4. Push a piece of Romex through the knockout. From the stairway, grab the end of the Romex by reaching into the hole you cut in step 1.
5. Wire the first 3-way switch in the existing electrical box with the existing wires and the new piece of Romex.
6. Pull the Romex from step 4 into the old work electrical box and install the box in the wall.
7. Wire the second 3-way switch with the Romex. Attach the switch to the box.

-- amateur

wdccruise: I wondered about that and tried googling for a wiring diagram, but got multiple results and wasn't sure which one was right for me.

Maybe this will help: https://www.youtube.com/watch?v=oaOCTAkKZhY

I thought modern codes require switches on the top and bottom of stairs. I would do what wdccruise says, but also figure out a way to run another piece of cable to the top of the stairs. Either eliminate the switch behind the door or wire a 4-way circuit.

MTVHike is correct. You must have switches at the top and the bottom (well, you might get away with a motion detector).

mtvhike and Ron Natalie: There ARE light switches top and bottom for the stair light. The problem is the position of the light switch for the basement lights after opening the door at the bottom of the stairs and entering the basement. I've edited my original post to make that clearer. Thx.

wdccruise: Thx for the link. I've given it to hubbie to study.

Thanks for the clarification. So you want to put a switch at the bottom of the stairs in addition to the one already there? Do you really need the old one on the other side of the wall (I can see that you might if you plan to close the door and turn the basement lights off)? To make things neater, you could either replace the box at the bottom of the stairs with a double box and put your new switch there. Or, you could replace the switch in that box with a double switch, although I don't know if double 3-ways exist.

mtvhike: I want to keep the switch at the bottom of the stairs that controls the stairway light. I want to add a switch that controls the basement light. And, I want the impossible of doing this without tearing up a lot of finished walls/ceilings. :-)

I think if you remove the plate for the switch at the bottom of the stairs and see if you can do. If the box is attached to a stud on one side, enlarge the mounting hole on the other side to accommodate a double box. I suggest you do the examination, then go to the hardware store to buy the box, making sure you get one which will fit. Is the wiring NM (Romex), or metallic?

Would it be easier to flip the door over so that it's hinged on the right? Would it solve the problem?

wOlley32: I suggested that but hubbie is reluctant, partly because it's a 9ft tall door and there would be a lot of trim to replace, cut-outs to fill in, etc.

wdccruise: This looks very promising. Thank you!

What about a light bulb socket or switch that has a sensor, so when you open the door it turns on?

Kate: Looked into that. Couldn't find one for a ceiling fixture. The light switch version would be around behind the door where it wouldn't see us and the door doesn't open wide enough to trip it either.

I’ll second the Lutron Caseta with Pico remote option. i have one in a similar situation, and and it works like a charm.

wdccruise: The Lutron Caseta idea was brilliant! We're all hooked up. The remote even works through the closed basement door. Thank you! And thanks to every one of you for taking an interest in my dilemma and sharing your knowledge.