I think providing free transportation would definitely encourage a reduction in drunk driving-related deaths. Many people choose to drive when they are too drunk because they have no other way to get home. However, it wouldn't prevent chronic alcoholics, who often drive because they don't feel they have an actual problem.
Yes, I think that free transportation would definitely eliminate drunk driving and it would help those who choose to drink get home safer. When people drink they lose control of thoughts and emotions, including making rationale decisions. Free modes of transportation would allow them to get home safer without having to worry about getting in an accident.