Such an elevator system might allow the formation of tight-knit safe and trusting communities on each floor, but I suspect that the common areas (lobby, and the elevator itself) would rapidly become as trashed as your stereotypical subway station.

My off-the-wall hypothesis is that poverty breeds crime, and crime breeds decay which itself breeds more crime (see: broken windows theory). Small tight-knit communities seem to, in some situations^ at least, put the brakes on this process. The tighter the community the more of a support net individuals have and the more obligation they feel towards their surroundings.

^ A counter-example that comes to mind are your stereotypical trailer parks and near-ghost-town mining towns across the rust-belt. A poor neighborhood that isn't dense isn't necessarily a wonderful place.