Yes, absolutely! I bought a gopro camera to do exactly that on an upcoming voyage. And some of the Fiona video footage was taken with a go pro.
I'm also planning to mount the go pro to a kite flying of the stern under sail - I've seen some amazing footage of boats under sail taken this way. But most of all I want to try to catch under water footage of dolphins riding the bow wave. I'm going to mount a waterproof goPro on the end of a pole, and attach the pole to the bowsprit with the camera 2 feet below the surface.
One of my most popular videos was of dolphins off Portugal... Fiona Sailing with Dolphins - YouTube
I'm thinking that if I could had also caught underwater video of those dolphins looking into the camera... it would have just been beyond amazing..
Thanks! I appreciate any advice!