When a father has visitation rights, but the child lives in NY and he lives in PR, who has to pay for the travel expenses?

The Father, since the son will be visiting him. Often the costs are split between the parents, but do not hold your breath on that one.

IF one parent moved out of state with the child, it is possible that court may hold them responsible if the other parent files for that.
But it is up to the terms of the child custody agreement if addressed.