Hi and thank you for your question.
Based on the information you have supplied, you have a fuel supply problem.The fact that your ski starts and runs when you supply fuel down the throats of the carbs speaks to that. It means you have enough spark at the correct time and sufficient compression to make the engine run. It takes 3 elements to make an engine run and the 3rd one is fuel. The carbs on a water craft have two sides. The fuel pump side and the fuel metering side. They also do not have a float bowl like a car or motorcycle have. There are two distinct probabilities of problems you might have right now. The first potential problem is the float needles could be stuck. If a ski has been sitting for extended periods of time, this is very common. If the float needle gets stuck in the needle seat, it does not allow fuel to flow into the fuel chamber and ultimately be delivered to the cylinders. The other highest probability behind the float needles being stuck would be clogged jets. In either event, you will need to take the carbs apart on the fuel metering side. That is the opposite side where the vacuum line attaches. To help, I am attaching a link to a diagram of the carb parts for you:
The needle I am referring to is reference #13 in the parts diagram.
The other items would be the jets are plugged or clogged. Pine Sol does an amazing job cleaning carbs so if they are looking like they are dirty, soak them in Pine Sol to clean them up,. This could take awhile so I suggest throwing them in the Pine Sol and checking them after the 30 minutes and see how they are doing. If necessary, let them soak longer until they look clean. Once they look clean, rinse thoroughly in clear water and re-assemble and re-install. If you can see through the jets, the needle valve opens when you press on the back side of the float arm, then go with full choke and try to start it.
Be aware of any other potential problems like perforated diaphragms while you are working on the carbs too.
If you need more help, let me know.
PS, if you need more help, please be more specific which 98 SL 650 you have, there are several models.