@bikingisligr welcome to JREF.
Hmm, I suppose it depends on the depth of their background check. I would expect a DWI to show up, especially since it usually results in an arrest, but would it throw much sand in your gears? I'm not sure what the statute of limitations is, but 12+ years is a long time and it may not matter.
Ironically, I have a friend who had an arrest in his past (I can't remember if charges were ever filed or if the case was just dropped, but he said he passed out drunk in a cab and awoke to a beating from the police). He was originally going to work in Korea and the immigration official decided to deny him, so he ended up going to Japan instead. Don't put all your eggs in one anecdote, but if the charge does rear its head and you get grilled, you could just say it was so long ago you had forgotten about it, since it was such a minor thing and you're a changed person, etc etc. I guess you could also look into sealing the record if that's possible.