Does it turn over? If so the starter is OK as is the starter solenoid. If it doesn't turn over then I'd try jumping the starter solenoid as a first step, since this is a common failure point (jump the 2 large leads, which are the positive lead to the starter). Look for loose connections to the starter solenoid when you do this, because that is also a common problem.
The engine has to be turning over to get spark, so once you get it turning over, pull a plug, and verify that there is still no spark. If there is still no spark with the engine turning over, post back with the engine type and I can help you check the charge and pulser coils.
I do not have a service manual for that engine, so voltages and wire colors may not be correct. If you can find a service manual it will give you the correct specs.
1st step would be to check the kill switch and fuses. Very common for kill switch wires to corrode or go bad. Just dissemble and bypass and see if the ski will start. If so get a new switch assembly.
2nd step is to check the plug wires. Very common for them to corrode/break. Take off the boots and check. Cut off end of terminal wire and reconnect if it looks bad.
3rd step is to check output of charge coil while turning engine over. I'm guessing voltage should be between 30-50 volts AC. You need to disconnect wires from the CDI to check it. If left connected to the CDI you will not get accurate reading. On later model skis it is brown/white wire. You check voltage between it and ground.
4th step is to check pulser coil while turning engine over. On later skis it is white/red wire. Disconnect from CDI and check voltage between this wire and ground. On later skis it should give a pulsing voltage of 4-5 volts AC (you may need an analog meter to see it).
If all above check out you can be pretty sure you have a bad CDI, which tends to fail fairly commonly on a ski this old.
Before I get into checking coils, I recommend as I did above that you check for loose/corroded connections and bad switches first, including plug wires and connections, since this is probably the most common problem for no spark. Also check connections to the rectifier/regulator assembly.
Charge coil = the coil that charges the capacitor in the CDI (I don't know why Yamaha calls it this. The coil that charges the battery is called the Lighting coil).
There is also what Yamaha calls an ignition coil that steps up the current from the CDI (capacitor) to the spark plugs, however I do not know of a way to test the ignition coil (you need a very sensitive ohm meter, and it will only give you a guess as to the ignition coil health). Basically after confirming that all the other components are OK then look for a spark to see if the ignition coil is working. The ignition coil is the only coil that is not in the generator box. It connects directly to the CDI then to the spark plugs. Look for bad connections at the ends that connect to the CDI and spark plugs. The ignition coil is a pretty simple step-up transformer, and there is not much to go wrong with it. Problems with connections to and from the ignition coil to the spark plugs are the most common type of problems associated with it.
The lighting coil is used to charge the battery and sends current to the rectifier/regulator then to the battery (but Yamaha calls it Lighting coil not to be confused with Charge coil). The Lighting coil sits right next to the charge coil inside the generator (you can ignore it for these tests).
All the coils are located in the generator assembly if you look on a parts diagram except the ignition coil. The charge coil will be one of the larger coils and the pulser will be smaller. As the engine turns over these coils generate current and send that current to the CDI or rect./regulator, etc.. Look at this parts diagram for where the coils are at: My Yamaha Prompt - Parts Catalog
You don't need to take the generator apart to test the coils. You can test everything where the wires tie into the CDI box from the generator.
How it works: The engine spins a magnet in the flywheel generating an electrical charge in the coils that are located in the generator (charge, lighting, and pulsor = basic electro-magnet principle). The Charge coil sends current to the CDI which is basically just a big capacitor which stores the charge (CDI = "Capacitor Discharge Ignition" box). Then the pulser coil also sends a much smaller current to the CDI. The pulser coil tells the CDI when to release the large charge stored in the capacitor to fire the spark (basically controls the timing of the spark). The charge leaves the CDI and is stepped up in voltage through the ignition coil before it reaches the spark plugs. Both the charge and pulser coils have to be working to get spark.
TESTING: On the 701 engine you disconnect the brown/white trace wire and ground wire from the CDI and connect your meter to these wires to check the charge coil. Then crank the engine to test the voltage output (be careful, since this can be pretty high voltage). I do not have a service manual for the 500 engine, so I don't know if this is the correct wire color for it or not. I also do not know what the correct values should be for your engine. On the 701 you would look for around 30-50 volts AC with the engine turning over (it will vary as to how good your battery and starter are). On your 500 engine if you are in this range I'm sure the coil is good. Also note you need a meter that has a DVA (direct voltage adapter) built in, or you can buy a DVA adapter for your meter. I test with a Peak detection meter: Peak Voltage/DVA. If you have a standard meter it is much cheaper to just buy the DVA adapter (or if you are electrically inclined you can make your own). I have heard that you can test the charge coil with a regular meter and multiply the value you get by 1.41 to get the DVA value. I've never tired it myself, but if your reading looks low with a standard meter, try it. This is because the voltage tends to jump around with this test. You are measuring the peak voltage with the DVA. That is why some meters will not be able to read this voltage. If you use a DVA adapter on a digital meter you must set voltage to DC on the meter even though you are measuring AC. The adapter has a rectifier that converts AC to DC.
To test the pulser coil you disconnect the white/red trace wire from the CDI on the 701 engine and ground and connect your meter to these 2 wires and crank the engine. You look for 4-5 volts from the pulser coil (analog meter is best). Again I don't know if this is correct wire color or voltage for the 500 engine........you need a service manual.
You may find people that will tell you to test resistance values for the coils, but I have found that resistance values can be misleading. I do not know what the values should be on the 500 engine. It is possible to have OK resistance readings and still not getting voltage output from the coil. You have to test the voltage output that I have outlined above to be sure you have a good or bad coil. If a coil is bad you can buy a replacement for fairly cheap.
If the coils check out OK and you still are getting no spark, then it is almost always a problem with the CDI. I don't know of a way to check the CDI other than swap it out for one that you know is working.
This is as simple as I can make it. If you are still lost, you are much better off taking it to a qualified mechanic and let them check it out for you.
After getting some emails, I realized that I made some errors and probably confused some with my previous post about testing coils in a waverunner. I have made some corrections to that post, which I hope removes some of my mistakes. If someone else sees a problem let me know.