Regular taxis will go almost the dodo bird.
There will always be a need for some taxis/hand transit type service where people are needed .
While insurance is a grey area answers will be found. Similar debates on how responsible owners/car makers of will be if accidents happen with all the driver assist technology now currently in use. Not pie in the sky stuff. Adaptive cruise/lane assist and such
The replacements in my opinion will be self driving vehicles of many types some privately owned others not.
check out one version used from town to town. The tech is not good enough for city service .
http://qz.com/507795/worlds-first-self-driving-public-shuttle-is-smart-efficient-and-adorable/
I note in Toronto the Transit commission is mad at a new Uber type service. Maybe Uber. They basically have several areas where Uber drivers meet Many drivers and offer people direct rides to prime areas in the financial district. My understanding it costs 7 bucks instead of transits 5. The cars/vehicles do not leave until full but My understanding is that is not a problem.
If I was a taxi driver I would polish up my Resume soon.