If they set up the corrupt system and everyone is expected to pay bribes, then the newly incoming passenger is not really the initiator, in that case the corruption is systemic, and i would put most blame on management to not make steps to eliminate it, but i agree that in this setup the bribees are more responsible than bribers, as they have more experience and information about the corruption in the system.
But let consider a slightly different case - a bureau (say building authority), with many twisted bureaucratic rules and procedures, where any request takes long to proceed and is often denied, but generally not corrupt. People just patiently do the necessary steps and accept it, or shrug off and go away. Then someone does not accept that and instead offers a bribe to get pass their request fast. In such case the initiator is clearly responsible for corruption, while the bureau is just responsible for general bureaucratic inefficiency.
And lets consider third case - driving license exam, and someone who fails it just bribes driving inspector to instead make it pass. In that case it is not even inefficiency, the license should be rejected and the briber just seeks unfair advantage. That is even more clear that initiator is more guilty, although the driving inspector has the power.
So, i would say that there are two kinds of corruptions - first, where briber wants something that should get without bribes, and second, where briber wants some unfair advantage.