Title is self explanatory really but I have been wondering if there was a specific fund for the ferryman, because the size and scale of some battles/wars that must have been a huge cost to all parties involved, would this have affected the war effort through lack of funds for other things? Or would it have come out of that soldiers pay like a death tax?
You might be interested in this earlier thread about this custom. It doesn't directly answers your question but it gives a context to the custom and when it emerged.
More relevant to your question is the idea that: 1) This custom was only really popularized in the classical period and wasn't universal, and 2) The amount to "pay" was a single obol, which was the lowest denomination of coinage, so it's doubtful that even if it was universal it would amount to a significant part of the war "budget" compared to many other things.