Toy with non-waterproof battery box: is this an issue? What will happen?

If I understand this right, you've got a electronic product that is intended to be used immersed in water, but it leaks and water can get onto the electronics. Seriously!!!?

In case you really are serious, yes, that is a problem. Even if the water only gets to the batteries, it can cause corrosion and will provide at least some leakage path for battery current, which will discharge the batteries prematurely. Pure water has pretty high resistivity, but it doesn't sound like you get to control the cleanliness of the water. Even a little salt, for example, will greatly increase conductivity.

There is little danger from the electric current since it will be largely inside the battery box, but again, seriously!? You are really thinking this might be OK somehow?

Apart from the fact the toy won't work for long, it makes it potentially very dangerous and absolutely not usable.

If *water gets to batteries, then they could be effectively shorted out (read low enough resistance to cause problems). There is a risk of rupture (or explosion/fire) which may cause the contents of the battery to leak into the water the toy is supposed to be used in. This also depends on the type of batteries used.

My view is that since it's a toy, however unlikely, you must assume that the worst case will happen.

*perfectly de-ionised water is not very conductive, however most water is well ionised.

For interest's sake, based on the comments below I took a glass of water and some salts, an AAA battery holder and hooked it up to my supply whilst immersed.
As expected, the "pure" water had very little conductivity at 3V, but with salts added a peak of ~0.3A was reached. Remember surface area is important with solutions, so it may be more with real batteries. With two copper surfaces of few cm wide/across 1A was reached easily. Not very scientific, but (hopefully) shows how the resistance can vary widely.

Anyway, here's a (bad, sorry) picture, note the bubbling from the negative electrode:

From experience with a weather station in the Scottish islands : it's not very dangerous because there isn't that much power from AAA batteries. Water won't short out batteries the same way a screwdriver would short out a car battery; the leakage would be milliamps rather than amps and heating is negligible.

Still, that will drain the batteries in hours or days. Worse (especially with salt water!) the battery powers spectacular electrolytic corrosion, dissolving copper wires, the battery casing, component leads and PCB tracks anywhere the water can reach, quickly rendering the thing useless and ... quite difficult to repair.