There are dozens of banks in Silicon Valley and for day-to-day personal banking they’re all pretty similar so you should pick one convenient to you. It’s important to consider that if you use an ATM (aka Cash Machine for you Brits) from a competing bank you will likely be charged a service fee of $2 to $4 dollars, so don’t pick a bank that is far away or every time to you need to take out cash you’ll get charged. This obviously becomes less important as more and more businesses accept payments via your smartphone.
Alternatively, there are some banks in the U.S. that are online only. Because they don't have any physical locations they allow you to use any bank’s ATM and then reimburse you for any service fees.
Note that each bank will have its own policies, such as minimum balance requirements. Make sure you understand any requirements and fees before you open an account.
All bank deposits in the U.S. are backed by the government up to USD $250,000.
Banks in Silicon Valley: