The financial services industry in Andorra is growing by leaps and bounds, too. It has always been sophisticated, and before the advent of the dominant Euro, Andorra naturally worked in a multi-currency environment with pesetas, francs, pounds, dollars, marks etc all being integral parts of the banking system. But back when we first visited years ago, banks were mainly interested in hiding cash for shopkeepers from Barcelona, while now they are offering some sophisticated private banking services. Needless to say, they have all the normal facilities like internet banking and credit cards.
What if you want to bank in Andorra, without living there? You can do that, and it is no longer necessary to hire a local maildrop either, as was previously the case. You can open accounts for offshore companies, too. The only thing is, you must show up at the bank in person. The only exception is if you can visit one of the foreign representations of the Andorran bank you choose. Several banks, particularly Banca Privada d’Andorra and Andbanc (the former Banc Agricol), are trying to capitalise on their success at home by opening offices in other countries. Credit Andorra has also just recently obtained a banking license in Panama (known as Banco Credit Andorra Panama S.A.)

Andorra has five banks, all but one of which are completely Andorran owned. Foreign ownership of banks is still discouraged… no longer impossible, but the small size of the local market has not been attractive enough for foreign banks to jump through the necessary hoops.
If you're interested in opening an account, you could just walk in to a bank off the street, and maybe get an account opened, but as likely as not you would be turned away. It’s always better to visit banks with a local introduction. Q Wealth Report’s Practical Offshore Banking Guide for example has more information on this and contact information for a local service company who can make the right introductions. If yoy speak Italian you may be interested in the guide "I Segreti della Banca Offshore". 

Links to Banks in Andorra: