According to the rules on the US embassy website, anyone who's been arrested (even if totally innocent) is ineligible for the Visa Waiver Program and must obtain a visa.
However, the form which travellers complete on the plane does not ask about arrests. It only asks about convictions for crimes of 'moral turpitude'. Your boyfriend can answer 'No' to this question, without lying.
US immigration officials have no direct access to UK criminal records (or to police records recording arrests). The UK does pass some information on but, unless your boyfriend was arrested in relation to suspected terrorism, drug smuggling or other cross-border crime, it won't include details of his arrest.
It's probably too late to apply for a visa anyway - the process can take 5 or 6 months - but he's got nothing to worry about if he simply fills in the visa waiver form.
Chris