Under the rules quoted on the US Embassy website, anyone who has ever been arrested (even if totally innocent) or convicted of an offence is ineligible to travel to the USA under the Visa Waiver Program (i.e. with an 'ESTA'):
http://london.usembassy.gov/vwp3.html
You're obliged to apply for a visa. That's a lengthy and tedious process, involving getting hold of a copy of your 'police record' (even if that's currently blank) and attending an interview in London.
The good news is that (alleged) ABH doesn't usually fall within the USA's definition of 'moral turpitude'. (Conviction for such an offence, which could be for something as simple as nicking a Mars bar, automatically places a lifetime ban on a person entering the USA. It's possible to seek an exemption but that process can take over a year).
Since 'moral turpitude' isn't involved you should receive the result of your visa application within a week or so of attending your interview, so the whole procedure shouldn't take longer than a month or two. (Remember that your other family members will also need to apply for visas, or ESTAs, as appropriate).
My guess is that your visa application will be approved but the only way to be certain is to complete the application process.
Application details:
http://london.usembassy.gov/b2.html
Chris