No, they didn't ask us to open the cooler. We just put on a luggage tag and away it went Ü
We didn't have any problems with using the cooler. Infact, we put it out in the hallway with our luggage at the end of the cruise....since we too had brought too much water with us. They just took it away and didn't say anything.
I don't think I would worry about it. I think as long as it is in some kind of holding bin (container) that won't leak any liquids if some should break open, it should be okay!