They sure do. The ones that can't or don't want to hire a personal driver or go everywhere by taxi anyway. Because driving yourself isn't much fun in a city and you need to be sober every time, too.
I'm skeptical. For nearly any definition of "rich", someone can afford to take an Uber/Lyft/taxi everywhere, at least.
I think the real reason your average rich person would take transit is because in some places, at some times of day, it's significantly faster than driving. But I do believe there's some -- probably fairly small -- subset of rich people who ride public transit simply because they prefer to.
