In the case of the Duggar's a few things I think contribute. A) they have so many children that proper supervision is not possible. The oldest kids are "in charge" of the younger, more vulnerable ones. This would be the case in any large family. You can't properly supervise large amounts of children and keep up with it all.
B) The Duggars (and I'm sure other large families based on things like the "quiverful movement") demonize sexuality and natural things like self-exploration. The boys in that religious sect are punished for touching their own bodies. Anyone who knows a pubescent hormonal boy knows that they need a natural, healthy outlet to express those emotions. If sex is demonized, and self exploration is a punishable offense and an affront to "God" then that lends itself to problems too, especially if the actual crimes Josh is accused of are perceived to be on the same threshold. It's ridiculous.
These delusional people mate like rabbits, but demonize sexuality. How is it possible to turn out normal? They are a living contradiction.
Most super huge families do tend to be religious as well and with extreme religiosity is a tendency to not teach the truth about sexuality and normal biological functions. It leads to confusion, self-loathing, feeling constant, chronic guilt that you are angering "God" and something is wrong with you etc. When you feel like filth for having normal, everyday feelings - it opens you up to giving into temptations. I mean if God already is angry and you're already a screwup who only needs to ask for forgiveness after every transgression - where is the motivation to develop a true, deep moral code?
So, lack of supervision, dimwitted parents who can't stop breeding, placing more value on the males over the females. (I have to wretch when they paint Josh to be a "victim" but not the girls) and just the overall lunacy of these people's belief system makes it ripe for sexual abuse and excuse making.