1. We might still have capitalism: The original series is actually pretty socialist - it's certainly evolved beyond capitalism. We're not (as far as I can remember) shown any actual exchange of money in the new film, but the product placements are pretty blatant so it's easy to tell that Nokia and (horrifyingly) Budweiser are going strong.
2. Most people will be white. Oh there will be POC. There's Captain Robau. And Uhura. And Sulu. So we haven't actually died out yet. We're just not the majority of the world's population or anything.
3. Despite the fact that people are zooming across the universe, fraternising with all manner of creature and barely notice aliens standing next to them at the bar counter, non-American, non-English accents will continue to be hilarious.
4. Voice recognition technology will be used, and miraculously all the alien types who might need to use it will have physically evolved in a manner that will enable them to do so. But not human Russians, because they talk funny.
5. Women who go to the bar to buy a drink will still have to contend with random arseholes. Random arseholes will go on to have successful careers by way of an old-boys-club-ish set of values - "I knew your dad", "You were very brave, there, when you punched out those colleagues", etc, and said women will have to work under them and it won't be awkward at all.
6. Appropriate clothing for women will consist of impractically short skirts.
7. Yo mama jokes will continue to be in use and effective.
8. School bullies will continue to exist. No one will actually do anything about this except maybe vaguely disapprove of it.
9. The Beastie Boys will continue to be awesome.
10. There will probably be sandwiches.
(Having said all this, I loved it. I was entirely uncritical while the film was actually playing, and plan to watch it again. "It makes my Id cum heaps all over". )
11. Humankind will still have not come up with a way to make childbirth less painful. (What is this "epidural" of which you speak?)