What would happen if water splashed on a 12v power supply when under load?
There is a reasonably good possibility the supply will be damaged, so you should just assume it will be.  Whatever is connected to the output might also get damaged due to over-voltage.  You can not count on a excessive input current condition such that a fuse would blow.

However, the worst issue is loss of isolation between the AC input and the DC output.  You have to basically assume that any part of the DC side is essentially connected to the hot side of the AC line.  That can kill or injure any person that happens to touch the supposedly safe DC and ground at the same time.

If you really think this scenario is a possibility, make sure to use a ground fault interrupting breaker on the AC not near where the water can splash.