As JThomas stated, check connections first. Check both battery cables and the red power cable going from the rear electrical box to the starter. If they still are all tight and it still wont turn over then you will need to experiment more.
First, we know that the switch and the solenoid are working because of the clicking noise so it should be an issue with the connections, the starter or the engine. The engine is the next easiest to test. Has the ski been submerged or the hull filled enough to get water in the engine? I have discovered, through personal experience, that if too much water gets into the engine it will prevent it from starting or even turning over (starter not powerful enough to overpower the high compression in cylinders if they are filled with water?) Take out the spark plugs, leaving the holes uncovered and put the spark plug leads on the grounding spikes on the rear electrical box. PUT THE SEAT BACK ON THE SKI because if this is the issue water/gas/oil will fly out of the holes and its not pretty. The seat will prevent the water from hitting you and you will still be able to hear when there is no more water coming out of the cylinders. If this works, simply reinstall plugs and wires and crank until the ski turns over. If not, you need to test the starter.
Remove the battery and the starter from the ski. Place the battery on a table and clamp the starter to the table. The starter will jump when tested if it is not clamped to the table and could cause injury. Run a cable from the negative battery terminal to one of the bolt holes in the front of the starter(ground) and attach another cable to the postive post of the starter BUT DO NOT CONNECT THE OTHER END OF THAT CABLE TO ANYTHING ELSE YET. Now to do the test, quickly and briefly complete the circuit by tapping the other end of the power wire against the postive battery post. If the starter is good then you will see it spin while the battery is completed, it will continue to spin while the circuit is complete. Do not let it spin long, you can over heat the starter and fry it. If teh starter doesnt spin then the starter is most likely bad and needs replaced/rebuilt.