Banking in the UAE

Customer service is not a word I would use when it comes to any bank in the UAE .  It is a hit and miss for most of them. 

Which bank you choose depends on your needs.  For example, if you have an HSBC account in the UK and want to be able to transfer to it without any issue, you would need to get an HSBC account in the UAE and use Global view.  Ditto for Citibank, standard chartered and so on.

As a rule of thumb, the cheaper banks with the most flexible product offerings are local banks.  However, at the other end of the spectrum, should you get into a bad situation e.g. loan defaults, you would not want to be fighting with a local bank.

Personally, when I was in the UAE, I had an HSBC account but got my credit card from Emirates NBD as they had the best offer.

The banking industry is regulated and generally secure.  Your choice of bank depends on your needs and accessibility.  Accessibility btw is not an issue in the UAE as ATM coverage is wide spread (and you can do free withdrawals from other bank ATMs if your account provides it), most banks have small branches in every other mall and payments/deposits can be done in machines everywhere.   Online banking is widely used.

I would stick to the bigger banks.  International would be HSBC and local would be Emirates NBD, ADIB, Mashreq etc.