I guess 'with' has a longer association than 'by'.
She came here with a bag.
She came here by a bus.
'by', I guess, is used as a means to perform a task. After the task is over, we let it go. It is not so with 'with'.
In your example, if better governance is achieved because of the improved campaign financing system then I guess 'by' is the correct usage. ( perhaps the implicit assumption is that one who financed would ask for better governance as a right)
If in the context, they are trying to say that, there should be a continuous financing for the parties even after the victory then 'with' would be a better option, but I don't guess the continuous financing would be legal.