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.
You would do well to incorporate an elbow or elbows, if possible into the part of the drain line that is behind the white PVC pipe, the metal pipe. The trouble with that would be is that old of pipe is easily damaged (broken) at the joints where the threaded sections are.
The reason I suggest this as the place to tie in the new extension is that by the time, I feel, if you simply replace the drain it will be so far away from the wall it will be a number of inches away from the wall it is against now. To remedy that, it would take so may elbows to get it back to the wall, I am afraid the drain would not drain readily or perhaps clog easily. So back at the reason why I mentioned taking it back and tying it into the farther in from the joint where it is tied in now. The place I refer to is at the wye, there is a small straight pipe there, perhaps there is a smooth part (no threads) to it to use a neoprene coupler there.
Best Answer
If there is already a trap underground, you won't need a trap under the sink - indeed if you added one, you will have major problems with draining with a double-trapped setup.