OK - I'll start with a warning: I'm no expert… but I do manage to fix broken things a lot :)
It's usually a good sign when it doesn't power up - signals that something is wrong with the power supply and not the more advanced circuitry. Power supplies can be fixed for just a few dollars and one trip to RadioShack etc.
Before you do anything, make sure you are using the correct external power supply (12V DC 200 mA negative center).
If you can confirm that a known to be functional DC power supply with the correct polarity is not working with the 707, then you have to open it to check the internal voltages.
Here is a schematic of just the power supply section:
umich.edu/~damont/TR707PowerSupply.tiff
Just follow the outside power jack to wherever this circuitry is in the TR. The boards will be marked with the component parts. With your red lead, check for +5V DC at the output of the voltage regulator (IC1) and check for -10V DC at the anode on the D2 diode (the side of the diode that does not have the line drawn on the actual diode). You need to put your black multimeter lead on some ground point to do this.
If you get these voltages… well, I probably can't help you much more.
If you don't get the voltages, first do a visual inspection and see if any of the solder points looked cracked. The power input jack on this is probably soldered to the board. It can crack off. Also look for cut traces etc. You could hit all the solder joints in this area with a soldering iron just to make sure. Sometimes you get lucky and that fixes things.
If nothing looks weird and if soldering doesn't help then work your way from the DC jack towards the points where the +5 and -10 should be and see where it vanishes. It's often just a diode or the voltage regulator that has burned out. Could also be a bad cap, but that is less likely, I think…
I hope this helps (and isn't too obvious).
best D