I know to keep a guitar/bass is not to play with wet hands. But, is it okay to play just with fretting hand a bit wet? I'm thinking about the neck, will it affect the tone?
Dirt, oil, and moisture accelerate the wear on your instrument’s strings and finish. Wet hands will make your strings wear out faster and may damage the finish of your neck. Moisture will also soften your calluses and skin, so you may end up with the same kinds of soreness and skin damage that beginners get. It’s a good idea to play only with clean, dry hands.
As others noted, a little sweat and grime is inevitable. I personally keep a cloth handy to touch up my guitar or dry my hands if I start to notice an accumulation while I’m playing. My main concerns are keeping my fingertips strong and avoiding long-term damage to the instrument (e.g., ruining the finish where my arm rests on the body).
On occasions, a gig may be really hot and humid.You will sweat, so your hands will be damp. This is life. However, once you've finished, it's always best to dry the neck/strings and if there's no paint finish, the body with a cloth.Strings are not designed to last for ever, and they are usually metal, so will revert to rust as soon as possible, thus drying them has to be a good idea. The tone will change when they start to corrode, although it won't be affected by the fact you're playing with damp/wet hands at the time.
So, no big deal, but why would you ?
Moisture will make your strings corrode slightly more quickly, but this is inevitable because of sweat and moisture in the air. You should try to avoid playing with wet hands if you can, but in actuality I doubt it would make a significant difference unless you're consistently playing with wet hands. If this is the case, you can buy coated strings which are protected (somewhat) from corrosion by a thin plastic coating, so they last significantly longer for players with sweaty hands.
I think it is also important to note that your fretting hand will probably cause less corrosion than if you are finger picking with wet hands, as your fretting hand is moving around between the strings and along the fret board, while your picking hand is hitting the strings repeatedly in the same location. Unless your hand is soaking wet, the moisture will not be enough to cause any damage or warping to the neck.