Hi Ana,
Im cautious to jump to a conclusion, and say it is one thing or another.
Water leaking all the way downstairs sounds like a lot of water to be
coming from splashing about. Any chance you can post a picture?
You can seal your grout and it will help to keep it from staining, caulking
will stop a direct flow of water. If it is pooling against the tile wall, and
there is no waterproofing underneath the tile it could amount to enough
water to travel downstairs i suppose, at least enough to be noticed.
I dont want to scare you but, have you checked the pipes on your
whirlpool for leaks? You mentioned it is new, but sometimes even new
ones leak.