Hi and thank you for your question.
If you have a new battery and the engine is spinning over, then we can look at a few other things, First, what condition do the spark plugs appear to be in? If they seem to have heavy deposits of carbon or are black and oily, it would be best to replace them.
The next thing to look for would be to see if the spark plugs are getting wet with gas while you are trying to start it. You do want them to be wet with gas, but not dripping. There is a point of diminishing returns if you get too much gas. Too much or not enough and it won't start.
Another thing to check is to make sure the ignition system is actually working by checking for spark. To do this, remove the spark plugs from the head, put them back in the spark plug caps and lay them on the head, but AWAY from the open spark plug holes. While observing the spark plugs at arm's length, crank the engine over and see if you see spark jumping between the side and center electrode. It should be a nice fat blue spark. If it is white or yellow, and appears to be acting more like a sparkler, then the spark plugs are no good and need to be replaced. That is what a fouled plug does when tested. A fouled plug will not spark when under the compression created inside the engine as you try to start or run it.
Last would be to check compression. You need a compression tester to perform that task. If you have one, go ahead and test all cylinders. If you don't have one, I would check the other items first and if all is well with them, borrow or buy a compression tester to test your engine,
If there is any question as to whether the spark plugs are usable or not, replace them. Relatively speaking, putting new plugs in your engine is the least expensive diagnostic tool and procedure you can perform.
IF you have any more questions, feel free to ask. Also, if you could provide the exact make model and year of your ski it would be beneficial as well.