Congrats on the offer! No, they shouldn't care about DUI's because:
1. A lot of folks have 'em. That would have to be a really high paid job to be able to filter out folks with DUIs.
2. They didn't ask for it. If they wanted to know - they'd ask.
3. Like onyx said, think of it in a context of a job description. Unless you are driving a corp vehicle, it shouldn't matter.
I can see if there were two identical candidates lined up, they might favor someone with no DUI, but you're already in the 'offer' stage.
As far as I know, the only jobs (outside of being a driver) that care about DUI and prev. use histories are jobs that require security clearances. Even then it matters how long ago that happened, the circumstances and of course the level of clearance required, as well as many other variables.
Don't sweat over it and again congrats!