I think the answer lies in people somehow stepping up to their responsibilities when becoming a parent.
It is just way too easy for one parent (and it is usually the man, though not always) to walk away. The courts may catch up with them eventually, but there are ways around the system.
I wish it was possible to force people to be responsible for 50% of EVERYTHING that a child needs. Whatever the relationship between the parents, once a child is born, EACH of the adults should have to provided 50%. Married couples may reach an agreement that they will trade, ie, if you work x number of hours outside the house, I will look after the child in return for the financial equivalent.
Then if something happens, both parents know their responsibilities and their rights.
Sadly, illness/unemployment/death/special needs children etc are far more 'normal' than most people believe. The number of families who just assume that they will have healthy kids, never become unemployed or too ill to work etc. I know how hard it is, but every adult really should be thinking ahead - putting money into savings, pensions etc. I do think that having welfare has made people a little blase about the possible long term effects of some of their decisions. Look at how angry people are at the idea of working beyond the age of 60. People see it as a right to retire and have a pension, when really it's not.
Having children is expensive, living is expensive, and every adult should be prepared for that.