A buzz that decreases when you touch the strings or other metal parts on your bass sounds like a grounding or shielding issue. It isn't necessarily entirely the fault of your bass, though.
Are you sure you're using a shielded instrument cable, not a speaker cable?
Some things to try... are there any lights on electronic dimmers, or fluorescent lights nearby? Turn them off, does it affect the buzz? If the amp has a 3-prong (grounded) plug, is it properly grounded? If it has an ungrounded plug, is it polarized (one prong wider than the other in the U.S.)? If not, try reversing it. There may be a "polarity" switch on the amp, try that? Try moving your amp to a different outlet, the farther away the better (different room, different building), does this affect the buzz? Can you borrow another instrument/amp to try in the location where you're having trouble?
Last time I had my basses in a music store trying out cabs, they both had an awful buzz, the salesperson told me I had a bad shielding problem. I think they just had dirty power there, I have no problem at home, and some minor problems in our practice space (and I'm not the only one to pick up some buzz there).
It doesn't sound like this is your problem... but pickups can also pick up hum from the amp's power transformer or other magnetic sources. This won't be affected by touching the strings. It will get louder as you approach the amp, and usually fade in and out if you stand in one spot and rotate your upper body, changing the orientation of the pickups without changing their position very much.