This is a popular theory with certain economists but has not been borne out by experiment. E.g. my country introduced pricing for prescription drugs (previously free) based on exactly that reasoning. Turns out overall usage barely changed (the very poorest stopped taking their prescriptions because they couldn't afford them, but they're a relatively small proportion of society) and administering the payment system cost far more than the money saved in reduced usage or collected in fees.
Alternatives don't just compete on price, but also things like convenience. In fact, even in the current system, it's damn hard (impossible?) to find a transportation alternative that's priced lower than a bus. Muni in SF costs $2 a ride (plus free transfers for a few hours), and the only cheaper transport is walking, which can't "go out of business". If I take a taxi, it's not because they're offering $1 fares, it's because it's faster, more direct, and more comfortable, and that advantage will only increase in a free-bus scenario (because presumably buses will be more crowded).
You can, however, use it. Which is a good thing (a social good!). Even more, transit capital is allocated by use, not by profit. More people using it should (in a non-stupid world) lead to more investment in the infrastructure, which has long term positive benefits for everyone (or, at least, has had them every time in the past in NYC).