Brooklynite here. In theory, I want to be behind taxi drivers. They're a local business owners with ties in the community, and I would rather support them than a Silicon Valley startup with a questionable stance on privacy and poor treatment of their workers.
In reality though, taxis need to fix a few issues:
1. I don't need to be able to hail a cab anywhere--I can't really hail Ubers, either. But if I call a cab, I'd like to have them show up within, say, 15 minutes. The reality is that the explicit promise given by dispatchers is usually an hour, and after an hour, the taxi often doesn't show up. And if I am in an inconvenient area, the time I most need to call a cab, that increases the chances of lateness or no-shows. This could be fixed by fining taxis for lateness or no-shows.
2. Auto-playing ads in taxis. The last thing I want after a long day is a screen yelling at me about some TV show I would rather have a root canal than watch. Unfortunately, it seems more likely that Uber will add ads than that taxis will get rid of them, and I don't see a way to prevent this. Advertising ruins everything.
3. Taxis that will actually take you somewhere inconvenient without a fight. When I lived in Flatbush, most cab rides started off with me having to threaten to call 311 to get them to take me home.
4. Racism. As a white person in a black neighborhood, I've watched black people try to flag cabs and almost universally the cabs just drive past empty. I've gotten in the habit of flagging cabs for black people, and even after the cab stops, sometimes when they realize that it's a black person getting in the cab instead of me, they drive off.