Should second trap under kitchen sink S-trap be there?
No, there should not ever be two traps on a single drain line.  

The point of the trap is to create a water barrier which prevents sewer gasses from coming up the empty pipe and into the house.

When you have two traps on the same line, you end up creating a kind of vapor block in the line which prevents it from draining properly.

What do you do?

Get yourself some 1 1/2" PVC (not CPVC, which is yellow, PVC which is white) with a 90 deg elbow, and also get two Fernco couplings (black plastic couplings with straps on each end that can be tightened) that will fit the pvc and iron drain pipe.

Use a metal blade on a sawz-all and cut out the p-trap in your basement. (Have a bucket handy!) Leave room to connect the Fernco coupling on both sides.  I'd cut the right hand side out just past the coupling.

Measure and cut the PVC so that with the 90 elbow the ends line up with the newly cut drain lines.  Glue the PVC together and confirm the fit, then connect to the old drains with the Fernco.

Viola - a proper drain line that will drain smoothly w/ no vapor lock.