I've rarely had a brainstorming session turn negative that didn't either have one of the participants doggedly pursuing an idea almost everyone else disliked or one that somebody present really disliked from a prior parallel case. Usually it only takes on a personal attack flavor if one of the people critiquing it actually starts personally attacking, or is perceived to be personally attacking, the person who came up with it.

The answer is pretty easy if it isn't family or work, stop brainstorming with the person with zero sensitivity or the person who is hyper-sensitive to criticism. The ideal brainstorming crew is 2-6 people including one or more random idea chuckers and one or more default Devil's Advocates, none of whom get touchy about criticism of an idea they've lobbed out or personal in the criticism they dish out. If you don't have a group like that, there's precious little to be gained anyway, so form such a group or don't bother doing it at all. If you do have someone who is really good at coming up with ideas but also very touchy about criticism of them, essentially someone you need but have to use kid gloves on, the Compliment Sandwich works pretty well, like "I love that idea but I'm not sure if it's practical, but you're our ideas guy, so maybe can you think of any way to deal with [the problem you've just raised]? I'm stumped."

It might help if you gave more details though.