Generally speaking, taxis around Bali are very safe. For extra security you can use and would recommend a ride app like GOJEK or Grab. You’ll know what your price is going to be and can track the journey.
The drivers in taxi cars are usually very careful, likely because traffic in Bali is crazy at the best of times and they’re looking to keep their car scratch or accident free. Taxi bikes on the other hand can be a real mix, some drive a bit wildly between cars and others drive super slow. If you get uncomfortable on the back of a bike, tell them to slow down – most will understand English enough for that.
I have heard of occasions for women, though, that their drivers have been a bit too flirtatious or “pervy” which can be concerning. I have not heard of any occasions where it has gone anywhere beyond that, so I would still consider it safe for women to use a taxi.
I would advise any woman to be very direct and set clear boundaries if the driver is being suggestive in any way. If it becomes uncomfortable, tell them to stop and get out of the taxi (or off the bike) and tell them to leave you alone immediately. Of course just make sure you’re in an area where you can get another taxi!
Generally I wouldn’t advise anyone to use random drivers on the street, often these can just be local people trying to earn some money on a Friday or Saturday night. They might and usually are well intentioned, but you never can be sure – so just keep your wits.