The central question is two-fold: How much of a shift would result from a given decrease in price? And how do we relate the (primarily:) monetary cost of making it free-to-ride with the (primarily:) non-monetary benefits of any given shift? The result of this question could give you an answer if making it free would be worthwhile.
Some thoughts:
I think decreasing the price per ride from e.g. 1 USD to 0 USD would make for a bigger shift in uptake than decreasing it from 2 USD to 1 USD. Not having to think about whether each single tour is worth the price of admission makes it a viable default way of getting around. This is just the usual flat rate argument that also applies to things like internet usage.
Making public transport free would invariably result not just in a shift towards it from other modes of transportation, it would also lead to an overall increase in mobility, which in terms of some resources reduced the gains in efficiency.
There's a valid argument that the efficiency of public transport is highly dependant on the amount of utilization: big buses and trains carrying single digit amounts of passengers can use up more energy than individual transportation. An increase in overall uptake would tend to reduce such problems since you'd get a small bus load full of people in cases where you'd have only a few now.
Obviously, free to ride public transportation is a particularly huge potential improvement for people who otherwise could not afford to get around. And since mobility is such an important part of life in modern society (minus us nerds who manage to leave the house only once per week), free-to-ride public transport has a massive impact in terms of social equalization.