typically, it would be responsibility of the manufacturer himself;
because, establishing the suitability/adequacy of container closure system (packaging) is part of the development process...
and sometimes established through simulated conditions and few instances through transport validation(actually shipping the product and evaluating the product at the end of journey);
having said that, in case of special storage conditions required for the product. viz., maintaining < 5deg C, it becomes responsibility of the transporting-agency. (covered through your technical/quality agreement)