1. Many Latam residents need/want/use US bank accounts for many reasons. Some don't trust their local banks (Argentina, Venezuela), some want to hide their assets from their govt to avoid taxes, some have real estate or want to invest in the US and having a local account makes everything easier.
2. A very flexible account in the US is very cheap. You usually need $500 to $2500 in a balance to operate fee free. You will need north of $250k and even north of $1m to open a private bank account in Switzerland or Liechtenstein. Also those accounts provide almost no flexibily and are normally used for long term investment rather than as transactional accounts (such as the one needed by the OP).
3. For the OP:
a. Do not use your ATM card at POS. You will get your account closed more sooner than later. Dont't ask me why, just have lots of friends that went through this experience.
b. Be prepared to show where your money is coming from (invoices, proof of services, etc). They may freeze your funds if you can't explain.
c. Try to receive regular payments and from the same entity. If you have a company and bill over 10k to more than one customer, avoid using the same account to receive them. Open an account for your company or one personal account per customer. Otherwise, you will find your account closed.
d. Open a virtual address at virtualpostmail.com. Use your real residence address when you open your account, but your virtual address for mailing purposes.
e. Make sure you don't do business inside the US nor hire noone in the US nor stay more than 180 days per year in the US. Fill the W8 regularly so you don't experience withholding taxes.
f. Stick to big banks that have experience dealing with non-residdent aliens. Almost every bank in Miami will help. HSBC USA is run by idiots, however if you have an account in your country they will open one for you in the US very easily. In most cases you can even avoid traveling to the US as they can receive the forms in any branch in the world. If you have more than US$100k in either your local or US account, you'll get Premier status and wont have forex fees. They will also give you a credit card.
g. Get a credit card. Citi, HSBC and BofA will give you credit cards. They will probably request that you create and use a non cancellable CD (certificate of deposit) as collateral. AMEX will also give you one if you already have one and tell them you are planning to move to the US. You can also get an AMEX dollar account for latin american residents at americanexpress.com/lac
h. Get a credit card.
i. Did I say you should get a credit card? Though they are not publicized, there are plenty of options.
j. First get an account wherever you can. Then, try to get ne fee-free.
k. Using your credit card at your home country will expose your international banking relationship.
Your country needs your taxes. Don't use this info to hide assets or avoid being taxed.