Usually, after a careful inspection of the circuitry, you would want to verify that all power supply voltages are within spec.
In something that old, you should probably either test the ESR (Equivalent Series Resistance) of every electrolytic capacitor or replace them. If you don't have an ESR meter, but do have a scope and a square/pulse generator (or a 555 timer circuit), you can try the method at **broken link removed** . It does work pretty well, even for in-circuit caps.
If it's not something easy like an electrolytic, you should get the service manual.
It's been quite a while since I've worked on scopes, so this might not be quite right, but here goes: The deflection circuits are often a sort-of "balanced" two-sided affair, with two identical-looking sets of components, which helps in diagnosing them. When a vetical position knob is turned away from the center position, the voltages in the two parts of the deflection circuits will change from equal to not equal, moving away from equal in opposite directions from each other, by about the same voltage difference. Then, where you find that is not happening, you are near the problem. You should be able to follow the wires or traces from the vertical position pots to find the related circuitry.
If BOTH channels are not working, that should be a huge clue and should narrow it down considerably. (And might just point to a hopefully-simpler power supply problem, as someone already mentioned.)
Good luck,
Tom