He'll have to cross his fingers but he'll probably be OK.
Officially, he must obtain a visa. Since the first stage of the process (getting hold of a copy of his criminal record from the police) can take up to 7 weeks, before you even apply for an interview (2 to 3 week wait for an appointment) and wait to hear the result (14 to 16 weeks is the minimum time quoted by the US embassy website), he's got no chance of getting in legally.
The Visa Waiver Program requires him to complete a form on the plane. This asks about convictions for 'crimes of moral turpitude'. Drink driving certainly doesn't fall into this category but I'm not sure about drugs offences. Anyway, he''ll have no choice but to answer 'No' on the form or he'll automatically be turned back at the arrival airport.
The US authorities don't have direct access to UK criminal records. Some information is passed on but, for security reasons, the exact nature of such information isn't known. It's extremely unlikely that information about drink driving would be included but it's less clear about drugs offences. It would seem fairly obvious that information about drug smugglers is shared. It would also seem quite likely that information about possession of drugs 'with intent to supply' would be passed on. What isn't clear, however, is just how far down the scale of drugs offences the cut-off point is, below which information is not routinely handed over.
This type of question comes up fairly often. The consensus of opinion seems to be that your boyfriend will probably be admitted but nobody is really in a position to offer any guarantees.
Chris