My first impression is that someone did a bad plumbing job and that only the cold water source was connected to the two sinks. To test this, turn off the water at the hot water tank. Test it upstairs by opening the hot water faucet. If nothing comes out, then the valve is safely closed. Now turn on the hot water in those basement sinks. If water comes out, the hot water is not even piped into the faucets, and the cold water is pumped into both sides. If this is the case you are going to have to locate the nearest hot water lead and pipe it to the sinks. While you are at it, you should install shutoff valves on each feed under the sinks.