Uber is safe. You can caricature the UberX service as hitching a ride with a stranger, but this stranger has his name, photo, and personal info on record -- and his GPS location is tracked and logged.
If you are concerned about your safety, I recommend that every time you step in a car, you text a friend:
Code:
Just got in an Uber. Here's my live location: (URL)
The link (available in app during each ride) will share the GPS location of your driver's phone, so even if your phone runs out of battery, your friend will know where you are.