This is a very important issue that should be answered by some important points.
Your bank should have a physical branch to deal frontally for the beginning of your relationship. Mostly they are under the inspections of the financial bureaus handled by the government. They should be insured against any financial collapse of the bank to insure your account. Check who is behind the bank, names, address, phones, PO BOX and web site. Check the services they offer, which ones are free of fees and the ones you are charged for. Does the bank require you to use direct deposit or not, because it may cause you problems if your money does not come from direct deposit or if you have switched jobs. All these problems should be taken in consideration. Most of these questions may be found on the banks web site.