I have a sink in the basement attached to the drain pipe for the unstairs kitchen sink. There is a vent pipe in the basement sink between the sink and the main drain of the upstairs drain/vent pipe. The basement vent stops at the floor joist. When the upstairs sink is drained after doing dishes the waste water will discharge into the basement sink. Why? Help!

Unbelievable! This is called a "local vent". It presents a direct open connection between your home and the sewer allowing sewer gas to enter your basement. Sewer gas is harmful to your families health and the methane content makes it potentially explosive. My advice would be to run it out the roof, revent it back to a dry vent or install a AAV, (Air Admittance Valve, see image.) At the very least cap it off. Now back to your problem.
You have a blockage downstream from the basement sink. The drain line will have to be snaked out. As a rule you must snake from the roof vent but the only thing that comes to mind would be to cut the basement vent down to size and snake from there. But get that vent looked after. Good luck, Tom

Speed, what should the installer of this "local vent" should've done originally, and did they take the easy way out in this case?

Speed, what should the installer of this "local vent" should've done originally, and did they take the easy way out in this case?
The open vent should have been run out the roof or revented back into a dry roof vent. The simplest thing jcilick can do is install AAV on the open pipe.To check out Air Admittance Valves click on;
http://www.studor.com/
Regards, Tom