I'm not sure I completely understand the original question, but I think it boils down to how the masts can be kept straight and vertical while the model's under construction. The means by which the model is fastened to the baseboard shouldn't have any effect on that - provided that (a) the hull is lined up right vertically, and (b) it isn't allowed to wiggle on the base. If you're worried about the latter, consider cutting a few wood wedges to the appropriate size and jamming them between the hull and the baseboard until the model's finished. (Maybe stick some felt to them so they don't damage the finish on the hull bottom.)
Schoonerbumm's idea for lining up the masts is interesting, and I don't see any reason why it shouldn't work, unless the masts are raked at varying angles. It sticks in my mind that the Cutty Sark's are - very slightly (i.e., the mizzenmast leans a little further aft than the mainmast, which leans a little further aft than the foremast). If I'm right on that point (maybe I'm not), the masts are slightly farther apart at their upper extremities than at their feet.
Personally, I've always found it easiest to install and rig the masts of a ship like that one section at a time. That is - step all three lower masts, set up their standing rigging, install the topmasts and install their standing rigging, etc. There's no substitute for the good ol' Mk. I Eyeball in lining up the masts, but various things can help. The most obvious is to hang a plumb line someplace where you can always see it, and use it periodically to line up the mast you're working on. You can do something similar to set the rake of the masts: mark the proper angle on a big piece of paper and hang it on the wall behind the model.
A modern, high-tech (at least by my standards) tool that can be of help in this job is a laser level - the kind that projects a bright red horizontal or vertical line on any object that's in front of it. (I've got one that I bought at Sears about five years ago for about $35; I think similar ones can be had much cheaper now. It's easy to justify to the household's Significant Other: a laser level is great for such jobs as hanging picture frames, plumbing garden stakes, etc.) Set the model on a table and the laser level directly in line with the front of it. (Whether the table is level doesn't matter - provided it's flat and the model and the laser level are both sitting solidly on it.)Turn on the laser and point it at the foremast. If the mast is straight and vertical, it will light up throughout its length. (If the mast and the laser line diverge, the mast is leaning. If the laser lights up one or more spots on the mast, the mast is crooked.)
All these tricks will help, but I repeat: there's no substitute for developing an eye for this sort of thing. Remember also that, unless you're going to operate the model in some extremely low-tolerance environment (what it would be I can't imagine), what matters most is how the finished product looks to the human eye. If it looks right, it is right.
Hope that helps a little. Good luck.