I'm pretty sure Bank of America still allows opening accounts with just a passport and a U.S. address for correspondence. But it also varies by state, and you need to show up in person at a branch to open it.
I'd be
very surprised if that's all it took these days. Banking rules since 9/11 have made it even harder. You have to have a SSN, or if not from USA, an ITIN or EIN, depending on the account type.
The only banks worth even trying it with are the truly international ones, like Citibank and Wells Fargo. A local bank is gonna tell you to fuck off, since they won't have a clue how to deal with this.
Moreover, as a foreigner, you don't want to get involved with anything to do with US banking if you can avoid it, because worst case, it might make you tax-liable in USA (or make you ineligible for foreign financial services (!) thanks to FATCA). That's a world of pain you're better off without. (As a US citizen, it's a world of pain I wish I could eliminate, but I can't.)