The 01 and early 02's did have a oil burning issue related to revised pistons. I would hope any of these that did have issues (my 01 never had any issues) would have been fixed by now. There are over 100 changes made between the 2000 and 2001 model years according to Suzuki ad copy from 2001 models. The basic engine is the same, but the 01 does have a stronger clutch and different carbs and all of them have PAIR (air injection for smog, previous models only had it if they were California models). Alternator, front brakes, forks, steering geometry, frame & rear shock, as well as the obvious things like bodywork is a quick highlight of the bigger changes. Many folks have complained of the 2nd gen bikes projector headlights being weak (faired "S" models only), but there are fixes for that in the FAQ. The H4 lights in the 1G and naked 2G bikes do cast stronger beams stock for stock though.
My son has a 97 and I've ridden it quite a few times. It rides fine and I don't really notice that much difference. The suspension is a little softer than mine was when it was stock - I think, that was a long time ago - but that could also be a function of the bike's age and mileage. Both his bike and mine have just over 50K miles. We just put a new clutch in his because it was slipping, but mine is still doing fine. However, we have no idea what kind of life it had under it's previous owners, and it has had at least two others.
To answer your question though, If you want to save some money, there's no reason to avoid a 1st gen B12... and one could argue, due to the piston issue, that you may even want to avoid a 2001. If it was my money, I'd buy the cleanest, newest bike I could afford.