Banking is very safe in the U.S. with this country being a major industrial nation. No matter the economic state, your funds are protected as long as the bank is FDIC insured. You will notice the sign for FDIC is all banking locations. If you do not see it, which should be a rare exception, do not trust the establishment and go elsewhere.
There are several major banks in Houston with numerous branches where one can conduct business. If there is no branch nearby, you will find an ATM (automatic teller machine) within a few blocks. Many businesses, such as pharmacies, gas stations, and supermarkets also have ATMs. Many supermarkets also have a small bank branch inside so that you can access tellers for extended hours or save a trip when shopping for groceries and needing to do bank business.
Following the attack on September 11th, 2001, regulations for foreigners to obtain a U.S. bank account have become more stringent. We will explain in detail more in the following sections but nowadays, you must show that you are a resident and there are different ways to prove that.