Except all your assumptions are wrong.

Vicarious punishment is the worst method of behavior deterrence. People won't stop fare dodging out of a small amount, not even a huge amount, or even if almost all of fare dodgers were caught. They will just become selective and apply different strategies to dodge the fare. It's a fallacy to think that the existence of a punishment for a crime automatically decreases said crime incidence in the whole population. We know this to not be true (looking at abortion statistics over here).

Police presence is a deterrence, only and exclusively in those areas and time frames when they are present, and even then only partially (people still commit crimes, even when the police is present). We know that there's no magical patrol coverage that deters crime completely. People will just do the crime, when and where the police isn't looking and get creative about it. Not even permanent surveillance deters crime completely or permanently (UK's mass surveillance didn't even put a dent on their crime rates).

It might be true that some people who dodge fares can pay them. But to assume that it is not an undue financial pressure on some individuals is misguided and cruel. We already know what it needs to be done to reduce fare dodging, increase the service quality. When the value of the public transport is regarded as better than what's charged, people are less likely to try to dodge the fare. But if the service is de-funded, installations dirty and derelict, slow, infrequent and on top of all of that the police is there to fuck you for fare dodging instead of making it safe for all users. Then people will consider it not worth their money. Adjacent to this, student, commuters, elder, children, job-seekers and disability subsidies on transport also increase people's perception of the quality of the service. This all together reduces fare dodging way, way more than any punishment.

The NY Metro actually did it once, they cleaned the graffiti, repaired stations, fixed lightning and subsidized certain groups. It dropped not just the fare dodging but all crime inside the metro, faster than the state's crime rate drop overall. But Giuliani shelved the project eventually because subway is for poor people and poor people don't deserve quality services, apparently. Again, the cruelty seems to be the point.