Oy gevalt!  What a beautiful dishwasher!  Congratulations!!!!

The water intake valve is at the rear of the cabinet that the dishwasher tub slides into - IIRC there should be kind of clip on the slides which move the tub back and forth that you can release to allow the dishwasher to slide all the way out.  But be careful - there is also a power cable that connects the motor for the dishwasher to a kind of power outlet also at the rear of the cabinet.  Don't pull the dishwasher tub out forcefully - ease it out and then reach inside to disconnect the power cable.  With the tub out, you'll see the water valve.  You may not be able to find something that fits perfectly but you can adapt a modern one.  

Have a look at the attached pictures from my poor, broken '59 Princess; hopefully what I said above will make sense.  I don't seem to have a photo of the water valve... sorry!

Colour me turquoise green with envy.... LOL 

Unfortunately, that gasket is probably shot and will need to be replaced.  It was a weak point for these dishwashers, apparently...  

The original GE part is NLA, but you should be able to rig something using closed-cell foam tape.   

There's an old-style solenoid valve that's attached to the back of the cabinet frame that the dishwasher slides into on it's drawer-style rails. It's located, usually, at the cross of the two metal braces in the back (on the LH side of fig. 11 in the NEW-4 picture below). It is likely that the solenoid mechanism is sticking open (I've had this problem with vintage washing machines; usually taking the solenoid apart and giving it a good but careful cleaning has done the trick). The problem here is that nothing in this vintage is modular. The solenoid is connected via copper pipe to the house plumbing and then another length of copper tubing runs up over the dishwasher to the front of the unit where the hot water is injected into the tub. You'll need tools, some patience and some plumbing ability. I imagine you could swap out the old solenoid with a newer one, but I would need a plumber to do that, I don't know about you. You might also want to be careful when you take the machine apart; one of the GE pull-outs we owned had large black asbestos pads insulation inside that cabinet, at least wear a mask.

Could you post some pictures of the inside of the machine and the back of the empty cabinet and some of the parts such as the inlet valve? It would help narrow down possibilities and would make our day.

Is this the one that's blue and white on the inside? That would be the color scheme for 1962 when they introduced their Drop-down door line. The first dishwasher our family ever owned was a 1961 TOL version of this with 4 cycles called a "Custom". In retrospect, it was the best dishwasher we ever had in terms of useful and meaningful cycles, capacity, speed and quiet; and we had a couple of great KitchenAids and a Maytag Jet Clean. Those others were most likely better cleaners, but in my family that wasn't an issue since Mom hand-washed every item before loading and got royally pissed-off if anybody tried to skip that step.

If you ever get tired of your machine, I have a good home for it where it can play outdoors and chase rabbits...