Usually you can feel a horse getting ready to plop down. I have been laid down on top of before while crossing a river. It didn't hurt much, as soon as he hit the river rocks he realized it wasn't a good idea and jumped back up. I however was left sitting on my bum up to my chin in running water. It was rather amusing, attempting to run after him as he's trotting away. Good thing I wasn't alone.
But you can let them play and keep them moving at the same time too. Just let them paw, ask him to take a few steps, paw, a few steps, etc. Coming from experience, I wouldn't let your horse stand for an extended period of time because he may get the idea that laying down would be super fun. He'll get plenty wet from crossing, so no need to feel like he won't have fun just walking across the water.