A while ago, I asked a Satmar Chassid who lives in my neighborhood about why women do not drive.
He explained that one of the reasons is that it is not considered derech tzni'ut - a way of modesty. He surmises that perhaps part of this origin was that when cars were developed, they had higher entrances - i.e., you had to step up (climb) into the car. When doing so, women had to raise their legs, thus revealing the lower part of their leg (and if wearing a loose skirt, perhaps, a higher area was also revealed.) Hence, perhaps, that concern lent itself to the original prohibition.
Another reason is that driving and cars' concerns was considered a "man's" role. Consider, for example, young boys and teenagers' interest in cars; stats showing few women drivers, in general for numerous decades until a spike around the 60's - 70's (Neither of my grandparents ever drove; my mom never drove; none of my aunts, except 1 of them drove - there's some consistency, I think); how few female auto mechanics / auto dealers, etc.
Also, a general consideration is that women should be at or near home and outside, minimally. Thus, it's deemed that driving a car is unnecessary.
As for being a passenger, they are passengers for "minimal" purposes, and it's almost always with the entire family, including the kids. When they attend weddings, often even the infants come with them, even to a friend's wedding.
There may also have been a security concern regarding sole women drivers. They are more likely to be harmed than men, and their own cars could be used as the very means or place of attack, as well.
For various reasons, not having women drive has become a long-established tradition that numerous Hasidic rebbe's enforce and the community follows the rebbe's policy.
It is debatable as to whether this would be "halacha" or minhag. It depends on your analysis. Personally, I would consider this "halacha" as there is a halacha to follow rules of modesty. If this is the rebbe's interpretation of what fits into these rules, then, for these Hasidim, it is their halacha.