This is a diagram of all of the components to the Ignition system on early (80s) Honda V4 motorcycles.

So you think you've got a dead ignitor. First let's try to prove if it's an ignitor or some other problem. Ignitors with problems can often be intermittent, so you have to actually catch it failing to determine which one is at fault, but an intermittent problem where pairs of cylinders seem to suddenly drop out makes the ignitors your chief suspect.
Have a look at the above diagram. The diagram shows all of the connections that make up the ignition system on Honda V4 Motorcycles. Most of the V4 engined Hondas use a separate ignitor for each cylinder pair. If an ignitor is the source of the problem, two cylinders will be affected while the other two are not. If your problem affects more than two cylinders, ignitors are probably not at fault unless you're unlucky enough to have two bad ones. If you're convinced the problem is the ignition, but it affects more than two cylinders, check the battery, ignition switch, kill switch, main fuse, and wiring harness grounds for poor connections.
(Repair hint #1: when checking the main fuse, actually take it out and hold it in your hand. Main fuses have a nasty habit of cracking in ways that are not easily visible, but will fall apart into pieces when you actually take them out.) 
(Repair hint #2: there's a ground loop from the harness that usually connects to the coil mounting brackets that can cause odd problems if not connected.)
There's a couple of ways in which the ignitors be checked.
First, note that one of the ignitors is used to run the tachometer. If the tach goes wonky when the bike misbehaves, the ignitor running the 1-3 cylinders is a prime suspect, however keep in mind that things other than the ignitor such as loose connections or a bad coil could also cause problems with the tach. On bikes equipped with a fuel pump, the 2-4 cylinder ignitor runs the fuel pump relay. The symptoms here would be the bike suddenly acting like two cylinders dropped out and then a short time later completely dying as the rear cylinders run out of gas, however if the ignitor just goes nuts and supplies intermittent or poorly timed sparks, the fuel pump will continue to operate.
Best thing to try in order to try to narrow down the problem is to swap the connectors to the ignitors and see if the symptoms move to the other pair of cylinders. If the problem moves when swapping the ignitors, you've probably found the problem.
You can also check for dead cylinders by _carefully_ checking to see if the front and rear exhaust pipes are hot or cold. I don't recommend trying to touch them, as the pipes can be extremely hot and will burn you before you can react to the heat. Rather, just try and get near them and feel for radiated heat. If you find cold pipes, try swapping where the ignitors plug in. If swapping the ignitors causes the other bank of cylinders to go dead, you've probably found the problem.
NOTE: In some of these bikes, one ignitor is physically larger than the other and may have a different colored connector, but the two units should still be interchangeable. The larger of the two units has a rev limiter built in but is otherwise functionally equivalent.
If you swap ignitor positions and suddenly everything runs normally, you either corrected a poor connection at the ignitor connector or the disturbing of the wiring has temporarily gotten an intermittent connection within the ignitor to work again. In either case, you probably want to check out the ignitor for solder problems detailed in the second bullet below or problems are likely to reappear in the future.
OK, now you've found a dead ignitor, now what? There's a few possibilities
of what the problem is:
First,
pop the end cap off of the ignitor and slide the rubber grommet that the
wires go through off the edge of it. The cap is not glued to the
case, but is snapped in place. Next you need to slide the PC board
out. Sometimes this is relatively easy, sometimes it's a real bitch.
The PC boards of the units I have opened were not glued in place, but they
had been covered with a sort of waxy material that was used to protect the
board's components from the elements. This waxy substance was on the
edge of the board as well has over the components and effectively glued it
in place. The ones I've pulled apart so far have come loose without
damage to the PC board. I use a small
jewelers screwdriver to carefully pry on one of the harness wires right at
the surface of the PC board and the side of the case. You need to be
careful to pry very near the PC board surface and attempt to not put any side
loading on the PC board itself or it might crack.

Once
the board is out, inspect the solder joints. The pictures to the
right are examples of what poor solder joints look like. If you see
what appears to be a ring around where the leads are soldered to the boards,
this is an indication of cracks that have developed in the solder
joint and it is well on its way to failing if it hasn't already.
One of the pictures shows the rings, the other shows a large fissure that
has developed and the wire is free to pivot around the solder joint.
It was a miracle this ignitor was working at all. Use a soldering
iron in the ballpark of 20 watts and resolder the joints using rosin core
solder. Proper technique is to heat the joint, then add the solder
after the joint has melted. You should end up with a nice shiny cone
at the joint and if this is done properly, it should only be necessary
to heat the joint for a couple of seconds. Extended exposure to the
iron could lift the trace or otherwise damage the board. Do not use
acid core solder or the acid from the flux will eventually eat away the
copper traces on the PC board.
Inspect the other parts of the board as well for any bad solder joints
and resolder/repair as necessary.
The
other possibility is that an electrical component has failed. While
I consider this the least likely failure scenario, it does happen and at
the moment there's not much you can do about it. The active components
used in the ignitor are either obsolete or custom and obsolete. The
driver transistor (the part on the aluminum heat sink) is the only major component
(other than some of the discretes) that looks like it would be possible to find
a replacement for. I haven't personally tried these, but the transistor
used appears to cross to an NTE2315 or BU806/BU807 type of transistor.
Ignition Troubleshooting Checklist:
All cylinders have problems?