PhD - How dependent are public universities on funding from the US Government? I had a conversation with a potential PI regarding the funding situation from a public university and he left me with the following - "There is a threat of funding cuts for science (due to Donald Trump's policies) but the department will fight to ensure that this doesn't affect students"

That sounds good but in the end, is there now a strong reason to opt for private universities to avoid issues with funding? I'm an international student picking between a very strong (top 10) public program and a top 25 ish private program. Most other decision factors are roughly similar for the two.
As Nate explained, these cuts aren't relevant to whether a school is public or private; the factors are much more complicated than that.  Public schools in many states have gotten serious funding cuts in recent years, but that's not a decision that's made by the president, but by the governor and legislature of those states.  You can research whether they've been doing so at the public institution you're looking at, but of course, the situation could easily change after the next election.  I think this is just too complicated to really try to game out how things will be impacted; I think probably the best source of information is talking to other students at the institutions and the precise financial details in your offer.