I was using my garbage disposal upstairs, everything seemed to be draining fine. I happened to come down to the basement and could hear water. I went into the basement bathroom and water was coming up through the sink, along with the bits of broccoli I put into the garbage disposal.
The problem you describe sounds like wet venting. The drain line for your kitchen sink is also serving to vent your basement sink. If a clog forms downstream of the basement sink, water from the kitchen or basement could back up to the lowest elevation drain. Chances are the basement sink was just tied into the kitchen drain line some years ago. Depending on local code, this was probably not a legal connection, but it was functional. I suspect you have a partial clog in that drain line, at or upstream of the main. Time for a snake. If you can, I recommend installing a cleanout at the basement sink drain or vent.