The UK site advises all non-visa nationals (e.g., Americans, Canadians, etc) to bring the same documents they would need to apply for a visa. And it is part of standing advice that visa nationals (China, India, etc) bring the same documents they used to apply for their visa. I have given similar advice to those travelling to the Schengen zone. Be prepared with everything, you never know when you will arouse an IO's radar.
This would include bank statements. But these are among the most sensitive documents that a person can carry because they have almost everything needed to commit identity theft. So carrying bank statements around the world invites risk and the question asks how the risk can be mitigated.
The answer is somewhat counter-intuitive until you think about it for a while...
Take a black felt-tip marker and blacken out the account number(s). Similarly for any payroll deposits, blacken out the remittance id's. Blacken out any other numbers and codes that 'leak' identity information. Bank statements do not normally show the birth date, but if it does blacken that out also. Importantly, leave your name and address clearly visible.
IO's must respect reasonableness, and no matter how hostile they may seem, they will invariably yield to a reasonable proposition. Imagine the following interview fragment...
- IO: May I see your bank statements?
- You: Yes, of course, here you are. They are in descending order by date.
- IO: (affecting great hostility and suspicion) Hey, the account numbers are blackened out! What are you trying to pull on me???
- You: The account numbers are blackened out because these are critically sensitive documents that invite the risk of identity theft
if lost or stolen. Taking steps to avoid identity theft is advice taken from the UK government's web site.
- IO: So how can I tell if these are real or not???
- You: You can see that these are original documents and that my name and address is clearly visible, and it will sync up with all of
the other documents I am carrying, including my passport. And what
further information could the account numbers provide beyond that?
In most cases they will not even challenge it because they will have seen lots of people doing this already. In truth, account numbers do not tell the IO anything and it is reasonable for an individual to take precautionary steps to avoid identity theft.
Personal note: I used to do this as standard practice for entry clearance applications (100's of visitor apps, plus maybe 150+ fiance/spouse visas) plus standing advice for Americans/Canadians travelling as non-visa nationals. No refusals or removals.
Also note: Just because you have attempted to insulate yourself from identity theft doesn't mean you will be landed. You can still provoke an IO's radar if you are a smart-ass or have a vague premise or an inconsistent premise or are wasted on drugs/alcohol or try to play cat-and-mouse or get caught in a lie or any of the other things people get bounced for.
If you are reticent, you can print out this question/answer and show it to them.