Public transportation in LA is "safe," unless she'll be taking the Blue Line to Long Beach, which goes through the most dangerous neighborhoods in Southern California: Watts, Compton, etc.
That said, getting around LA without a car is not a viable option, except for those who are too poor to afford a car. It takes hours upon hours to take what would otherwise be a simple trip. From Burbank, it'll probably take an hour to get to Culver City, two hours to get to Santa Monica, and so on. It's nothing like New York, where you can hop on a subway anytime you want, and you'll get where you need to go in 15 minutes.
If your daughter really doesn't want to take a cross-country drive, she should check with the company to discuss her options, which may include a company car. If that's not an option, perhaps she can sell her car in New York and buy/lease another one in LA, depending on how much money she'd get for the old car.