The official rule is that if you have ever been arrested or convicted of anything you need to apply for a VISA.
However, as has been discussed many times, and proof provided by the UK police forces, the UK do not share information with the US. This means that unless the crime was serious enough to put him on the international watch or no-fly list, then they would have no idea about his crime, and they are asking you to volunteer this information. BUT, if you choose not to disclose and for some reason the US are aware of him as a criminal entity, he could find himself deported straight back on the next flight after landing in the US and trying to clear customs.
Having an approved ESTA does not guarantee admission to the US, that's down to the homeland security guard at the point of entry to determine 'based on the information available to the officer at the time' i.e. valid travel documents and no little red flags on his screen when he scans the passport!