Can I open a bank account with U.S. banks while visiting on a B1/B2 visa?

Yes you can. Very easily. The bank will require your passport (with a valid Visa), they might ask for some secondary form of identification (driver's license, other credit cards, etc.) They will ask you for source of funds, and that's about it.

You will need to provide an address (cousin? family member residing in the US? give their address). Whole process takes about an hour.