Does the noise change (get better or worse) when you touch the strings, the bridge, or the metal barrel of the plug?
Everything makes noise, and damn near everything picks up noise from the world around it. As mentioned above, pickups in guitars and basses are especially bad because they are pretty much specifically built to be sensitive to changes in the EM field around them - they are antennas. Worse yet, they are generally very high impedance internally and connected to very high impedance at the other end of the cable, which is technical speak for inherently noisy.
You can't completely eliminate that noise. The best you can hope for is to optimize the signal-to-noise ratio so that the sound you want to hear (the strings in this case) is enough louder than the noise that the noise is not a real issue. How much louder is enough louder depends on all kinds of things, including your performance dynamics, any compression/distortion downstream, and your own tastes.
Usually the biggest source of noise that we get from a guitar is our own bodies. Actually, I think our bodies tend to concentrate and focus the noise from the universe around us, and just happen to be usually be really close to the pickups, but... Most guitars include a string/bridge ground meant to alleviate some of that by "shorting it to ground" when we're actually touching the strings - (hopefully) most of the time we're actually playing. This is why I asked my initial questions. If that bridge ground is faulty, you can have more noise than you might otherwise. That's usually an easy fix, but not always.
The other thing you can try is to shield the pickup and control cavities in the instrument. There are various types of copper foil sold specifically for this purpose, but I always just use heavy duty aluminum foil like you'd use in your kitchen attached with spray adhesive. This needs to be connected one way or another to a good "ground point" in the circuit, which is again sometimes easy and sometimes takes a little work.
Of course you can try switching to humbuckers from single coils. There are also "noiseless" pickups out there (which can't actually be completely noiseless, just a lot less noisy), and Low-Z active pickups followed by a preamp can sometimes be quieter than High-Z passives. All of these options will change your overall tone, and usually require some fairly serious surgery to achieve.
