I put in new recessed lights as a part of a basement remodel. Everything in on new circuits so once I had the new lights up I removed the old ones. However I cannot remove the old wire for the old lights as it comes from upstairs and I cannot see where it is connected (without demoing upstairs which I really do not want to do. Right now, I have the end wrapped in electrical tape and put the whole thing in a closed electrical box attached to a floor joist above. Several items upstairs are on the same circuit, so turning the whole circuit off is not an option. Would it be OK to leave like this, in the box, as long as I cut a hole in the ceiling for access?

Yes, but Wire nuts are better. Mount the box flush with the ceiling, using an old work box, then buy a white hole cover at HD, screws onto the box, paint it and looks pretty good.

Thanks for your response. So wire nuts on each individual wire, then wrap them up in tape?

Yes. Wire nut and tape.

Chuck