Start gentle and cheap. I'd hit it with 91% rubbing alcohol and a stiff brush of the type used to remove soldering flux. Thin metal tube squished onto 3/8" long black bristles about 3/8" long. Go for the stronger (91%) isopropyl alcohol vs the more normal 70%. If you can gather any crud together on the board, try to remove it with some Q-tips.
I am assuming that *you* are assuming that the solder joints are good and that the amp works (or you are ready to plug it in and see if it works...the lamp limiter is always a good bet on the first try)
DeOxit is the preferred item for cleaning pots, but it is expensive. I've had almost as good luck with regular Freon contact cleaner spray which is much cheaper. I would do just the board, first. See where you are in terms of go/no go.
It may be that you have to pay special attention to the 1/4" jacks. Treatment depends upon whether they are the black plastic enclosed type or open Switchcraft ones.
ed: I respectfully disagree w/Sluckey. It doesn't look that bad to me. It looks like atmospheric corrosion, not underwater corrosion. Yes, it's corroded, but inside all those wires the connections appear sound, at least as far as what you've pictured.
Tranny lamination rust = forget it until you assess further. Usually means nothing. But *do* use the lamp limiter on initial checkout in case you have something that's a catastrophic short.