Why does my sink's strainer bowl connection leak?
If you see a leak as you describe then there is a bad seal in the slip joint between the black ABS plastic fitting and the tail pipe that slides inside the black pipe. The seal normally consists of a rubber, silicone or neoprene washer that fits over the tail piece and under the plastic compression nut as shown here:



In normal installation the compression nut slides onto the tail piece then the compression washer and then the tail piece is inserted into the plastic pipe. Once the slip fit length is adjusted the compression nut is slid down and threaded to the top of the ABS fitting.

Old compression washers can get hard, dry and crack. When this happens the seal is broken allowing for a leak to occur. Your probable best fix is to replace the compression washers with new flexible ones. Consider replacing the seals on both sides at the same time and the seal rings where the tail pieces attach to the sink drainer baskets. (Note that these upper seals are a different style/size of washer). When the piping assembly is apart clean all the surfaces of the seal areas so that there is no grit, grime or corrosion that could compromise the seal. 

And...No. Plumbers putty is not a suitable product to fix this type of leak.