Also, how fucking huge would a basement have to be in order for someone to live in?
If you have a ranch style house (single story), then in most cases the basement is just as big as the rest of the house. A 1000 square foot ranch house often comes with a 1000 square foot basement. Unfinished big basements are cheaper to build than unfinished small basements. This is because you have to put the foundation in below the frost line, so the basement in general has to be at least as big as the first floor of the house. You could theoretically back fill in to make the basement smaller, but that just costs money to make your house smaller, and very few people would actually pay extra money for a smaller home. In other words, basements are usually huge.
Then it comes down to finished vs. unfinished. Many homes are now built as cheaply as possible. That means basements are cement floors, cement or brick walls, and open ceilings. But finishing a basement with carpet and drywall only adds a few percent to the cost of the home. So, many homes also have finished basements. Like others have said, the finishing is often as nice than the rest of the house (or possibly better if you consider that the basement is where the movie theater, game room, and wine cellar tend to be built).
As for water, you either deal with it flooding on occasion (as a small child, I lived in a house that got 3 to 4 inches of water every spring in the basement). Or you put in a sump pump that turns on as needed to prevent the flooding. The house that I own right now has a basement with a hole for a sump pump, but no sump pump is installed. That is because my house is near the top of a low rising hill and the basement hasn't seen a drop of water in 16 years.
I lived in my parents basement during summers in college. It was free and 80 minutes drive closer to my girlfriend. My brother lived there for 6 months after college while saving up money for a down payment for a house. Our parents had a 2000 square foot ranch house with another 2000 square feet of space in the basement (our state doesn't allow basement rooms to be considered in the house's square footage). The basement had a kitchen, three bedrooms (two of which were legal since they had windows big enough to walk out during a fire), 1.5 baths, a room for pool table/darts, large seating areas for TV/games, and most was nicely finished. Only about 500 square feet of the basement was unfinished utility room / storage room area. Actually, you could say we lived there during elementary school through high school too.
Finally: you are missing the point. It is just a phrase, it doesn't mean that the adult child actually lives in the basement. It means that the adult child lives with the parents, regardless of where in the home the child spends most of his/her time. Even if you have no basement, it is still called living in your parents basement.