The first thing I would check is to make sure the ground wire to the bridge is connected properly. If it is not, you will get the symptom that you describe. Touching the strings should not make much difference in noise level if the strings are properly grounded.
I just recently had that same problem on a LP, the ground wire was making contact, but not very well. The noise was not all that bad, but I knew something was wrong since it would disappear when touching the strings. I was measuring a few ohms resistance between the bridge and the pot casings - not good. I had to remove the tail piece bushing, and re-do the wire in there - on guitars like that, its just the bare wire being in contact with the bushing, held in place due to the tight fit into the wood; over time the bare wire can oxidize and lose connectivity. On trem equipped guitars, the wire is soldered to the spring claw. Not sure how it is connected to tele or hardtail strat type bridges.