Douglas, I'm glad you posted this thread. I've never had the steadiest of hands and this has been a problem with learning watchmaking. My dominant right hand in particular has a bit of a tremor. As with others, I use my non-dominant hand to steady my dominant hand. I enjoy coffee, but that's obviously not helpful in this situation. I've also noticed that my hands are steadier after I've had a drink or two, but I wouldn't drink just for that. I try to be creative when dealing with really small parts. Rodico on the end of a piece of pegwood is one of the best tools you'll ever make.
When I was having trouble oiling and inserting a parashock cap jewel into a movement I came up with this Rodico Hershey's Kiss
Not only did it create a stable platform and free up my left hand when placing the oil, it also made installing it into the movement much easier. I just turned it over, placed it above the jewel hole and gently knocked it off with my tweezers.
To me, Rodico is a miracle product. So many uses. Most people say such and such is "the best thing since sliced bread." I say it's the best thing since Rodico. Of course almost no one knows what I'm talking about when I say that. :D
Another thing that helps me is to take a couple of deep breaths and let my body completely relax before starting. It can be very stressful to try to manipulate tiny, tiny parts that are easily lost or damaged. I've seen more than one comparison of watchmaking to bomb disposal work. Though admittedly the consequences of a mistake are far more serious for an EOD specialist than for a hobby watch repairer.
Don