The police are correct with regard to the rules but not necessarily with regard to the procedures.
The US Embassy makes it quite clear that anyone who has ever been arrested (even if they were completely innocent of any wrongdoing) is ineligible to enter the USA under the Visa Waiver Program and must apply for a visa. The application procedure is extremely lengthy, taking at least 5 months (and possibly longer). During much of that time, applicants are effectively barred from any foreign travel because their passport will be held at the US embassy.
For details of the visa application process, see my post here:
http://www.theanswerbank.co.uk/Travel/Question 289769.html
However, the police are only partly correct when they state that a check is done on all entrants to the USA to see if they've ever been in any kind of trouble. US immigration officers do check everyone's details on their computer database and, ideally, they would like to know about even the smallest misdemeanours. However, they have no direct access to UK criminal records and their database only includes the information which has been passed on to them by the UK authorities. For security reasons, neither UK or US officials will reveal exactly what data is passed on but, as long as your caution wasn't for something like possession of firearms or explosives, it's extremely unlikely that the information will be on the US immigration database.
Several people have posted, here on AB, to say that they've got into the USA by simply lying on the Visa Waiver form. (This includes a recent post from someone with Class 'B' drug convictions). You wouldn't even need to lie because the form doesn't ask about whether you've ever been arrested; it only requires you declare convictions for crimes of moral turpitude.
Chris