In my job I often need to check if people have the right to work in this country. The minimum I would expect to see is an EU passport plus proof of NI number. If the person is from outside the EU then I would expect to see their passport, proof of their NI number plus a letter of authority from the Home Office stating that they have the right to work / working visa (or equivalent).
If you have copies of all this documentation then you should be covered if it turns out that something is wrong. I imagine it is easy for people to get hold of fake letters from the Home Office stating they have the right to work, and I don't know what the employer's position would be if it subsequently turned out that was a fake. I don't think it would be realistic to phone the Home Office every time you employed someone who provided these documents.
Surely the NI card is hardest to fake, in that if they provide a fake NI number that is picked up the first time you register them with thevtax office as a new employee? And you wouldn't employ someone who didn't provide proof of their NI number would you? Unless of course he was employing her cash in hand, which is a resigning matter in uts own right....