If your (UK) business is in an area where you have specific regulations to follow for anti-money laundering, and you do, you will already know the answer. (If it is not in the UK, you may have to find out if dealing with UK-registered Bitstamp means you'll have to start complying with UK law and its requirements like KYC: Know Your Customer.) Otherwise, the first answer is likely no and the follow-up questions are not applicable.
Commonly, KYC includes requiring a scan or photocopy of customer's photoid (passport or driving permit) and
of a proof of address (utility bill), and checking that they are not on a list of people the UK does not allow you to do business with.