I got a visa for my trip last year - NTdog will love this - only because of a stupid 'drunk and disorderly' years ago.
I know, I know, shouldn't have bothered, but like everyone else, I was worried as the kids would be with me etc.
As posted above, I was nervous about being asked 'why the visa' and took the Subject Access statement with me.
Border Control took the 4 passports, and of course, mine went through last.....he never so much as batted an eyelid at the visa. No extra questions, awkward pauses, etc.
Other than the usual where are you staying, how long etc, that was it.
You'll be fine.
I'd suggest taking the SA paperwork in your hand luggage, just to produce if you did get asked anything awkward, but CBP know that the US embassy have already done that side of things.
The irony is, it's guys like us who apply for visa's, and the ones who've done time probably just stroll through using the VW, but there you go.