If you open up your generator compartment and look inside there will be a nameplate on the Onan generator. This will tell you the model number of your generator.
Some newer Onan generators have a display that will generate error codes that can help diagnose problems.
The first thing to confirm is are you getting fuel. The Onan generators have an external fuel pump. The easiest way to test this is to hook a clear rubber tube up to the outlet of the fuel pump and stick the other end in a gas can. When you try and start the generator you should see a flow of fuel. Alternatively you can hook the fuel pump up to 12 volts and make it run that way.
Be careful here as you are playing with an open can of gasoline and could be making sparks. Keep a fire extinguisher handy.
Another thing you can do without cracking anything open is just try to start the generator half a dozen times. You should smell gasoline from the exhaust if the engine is getting gas.
You can also pull one of the 2 spark plugs (the front one is usually easier to get at) and see if it is wet. If it's we and smells like gas, you are getting gas.
Next check for spark. You can get cheap inline spark testers like this for less than $10:
https://www.amazon.com/ENA-Universal-Automotive-Motorcycle-Fool-Proof/dp/B07GHBNFKZ
Beware it can be hard to see the light bulb light up in direct sunlight. But it's super easy - disconnect the spark plug wire from the spark plug, and stick this tester in the middle. Now the light bulb lights up as electricity goes to the spark plug. Ideally you'd test this on both spark plugs but it's hard to get at the one on the back side of the generator. Generally if you're getting spark up front then you are probably getting spark in the back.
The most common problem with these generators is the fuel pump quits working or the carb gets gummed up from not being run often enough. It is critical to run these generators
every month under load for at least 30 minutes. The heat and vibration and flushing with fresh gasoline helps keep the inside of the carburetor clean. This is especially true with today's ethanol gasoline. If your carb sits you can end up with condensation water inside the carb and things start growing in there. Varnish, gum, debris, mold and gunk clogs the tiny passages inside the carb and require a rebuild or new carb to fix. "Mechanic in a can" fixes like Seafoam, etc., seldom work.
Also check your oil level. As others said, some Onan generators have will shut down if they do not detect oil pressure.
If your control board is suspect, Flight Systems will diagnose it for $20 and put the price of the test towards a new board if you buy one from them.
1990 Winnebago Warrior. "She may not look like much but she's got it where it counts!"