Hm, the Core in Cave Story doesn't strike me as something that exactly gets "overheated" and is in need of cooling. Now, the core in Undertale, that's a different story, since it runs on geothermal energy, and relies on those ice cubes from Snowdin town to keep it cool.
While it is curious that the water level in the Core room keeps rising, this doesn't seem to correlate with any sort of change in color that indicates an overheating of the Core. And the Undead Core at the end of the game is able to survive just fine for a prolonged period of time without any water, and things don't get unstable until you destroy it, so I would rule out the idea of that water being used to cool down the Core.