It's kind of a standard answer around here, but Beaune is a charming town that makes a great base in the heart of wine country.
I don't know what your experience will be like in the off season, but I am sure some vineyards will be open for tastings. (The vines themselves will just be gnarled brown stumpy things, though.) There are also several tasting opportunities in Beaune.
I really enjoyed a tasting lunch in Aloxe Corton, just north of Beaune. You get some traditional Burgundian fare and, depending on your budget, four good to six great wines.
I was also pleasantly surprised by the white wines around Chablis, though this is a bit of a drive from Beaune. (The wines knocked my socks off, actually, so I'd say try some whether you go there or not.)
There are also worthwhile things to see and do in Burgundy other than tasting wine, so you and yours might want to give some thought to that. Have a great trip!