The first one is (almost) correct, and while I can think of a much more limited example where the second one would work (see below), it is incorrect as you have used it.
I say the first one is almost correct because it isn't "put stop to...," but rather, "put a stop to..."
So, to use your example, "Will the government put a stop to all those criminal activities?"
The only example I can think of where "put a stop ON" would be used is in reference to a non-cash payment that has been issued and subsequently withdrawn: "I paid for the tickets, but decided to put a stop on my check when I didn't receive them."
This would mean that the buyer sent the check to the seller, but then called his/her bank and had them block the receipt of this check by the seller when the buyer did not receive the tickets as promised from the seller.