However the US authorities are actually only really interested in convictions relating to drugs, and to offences of 'moral turpitude'. People who 'forget' about their arrests and convictions, and a apply for an ESTA (in order to enter the USA under the Visa Waiver Program) , have to answer this question:
"Have you ever been arrested or convicted for an offense or crime involving moral turpitude or a violation related to a controlled substance; or have been arrested or convicted for two or more offenses for which the aggregate sentence to confinement was five years or more; or have been a controlled substance trafficker; or are you seeking entry to engage in criminal or immoral activities?"
https://esta.cbp.dhs.gov/
An offence under the Dangerous Dogs Act does not appear to fall under 'moral turpitude':
http://www.faqs.org/qa/qa-8222.html
So the answer 'No' could truthfully be given to that question.
The US authorities have no direct access to UK criminal records, so many (most?) people with criminal convictions 'chance it' and try to enter under the Visa Waiver Program. Most probably get in without any problems but if, for example, the US immigration service erroneously suspected that the traveller might be a terrorist or carrying drugs, they could ask their British colleagues to run a check. If that check showed that the traveller had lied about not having any criminal convictions he would be automatically barred from entering the USA.
Chris