I have a steel washer tray under my second floor washer. I don't notice to be any louder than the same units without a tray downstairs. Regardless of whether it makes it noisier, you need the tray and a drain just in case.

They were required by building code in my area for non-lowest-level installations until about 8 years ago, after which machines using a certain low amount of water (basically front-loaders and maybe a few HE top-load machines) became excempt. It's unlikely a machine failure would dump a huge amount of water on your floor so I wouldn't bother with a tray; a burst intake hose is a bigger threat (but also is for faucets and such). Use braided burst-resistant hoses if possible.