Hi John,
The best solution is the one mentioned above, tightening the nut. If that works, it is the best solution because it's free, easy, and almost instantaneous! But, if it doesn't work, there are other solutions. The problem of the creeping tailstock has been around forever. It was there on my first Jet mini, and is there to a lesser extent on my current Jet mini VS. A friend of mine decided to make a bunch of square plates to go on the underside of the ways, to replace the round one that comes with the lathe. I bought one, and it works. I have not kept in touch with him, so I don't know if he still does that. His name is Keith Outten. At that time he ran the Sawmill Creek forum - not even sure he still does that either. I know, what kind of friend am I? ;-)
One thing you could try is remove the round plate - it almost looks like a washer - and glue a thin rubber pad to the side that would come in contact with the underside of the lathe bed. I have no idea if this would really work, but I also don't see why it wouldn't, either. And again, it's cheap and easy; and the worst thing that would happen is it won't work, in which case you'd still need to get that other part.
But I'd try tightening the nut first. If that works, you've got it made!
Scott.