I don't necessarily agree with that line being parroted out again and again (what's the point...people are already in the dire and i imagine uncomfortable situation of needing benefits, saying 'you should have planned for x' won't change that), however I do think there is a lot that parents can and should do to protect both their own and their children's futures and reduce the risk they will end up on benefits. This is personally what I did (I'm sure it doesn't cover all eventualities though and I fully admit I have been incredibly lucky, I'm certainly not saying only people who can do the below should have children):
Didn't conceive children until I could afford them and had the below outlined in place (abstained...tmi!)
Married to DH to ensure some (although limited) legal protection
Owned own property (with relatively small mortgage) and back up property which are let out (held individually in own names by DH and I so we each have our own) although I appreciate this will be beyond the means of most. Basically as secure as possible housing and asset base and not over stretching on mortgage/rent etc.
Savings for a rainy day in place (would recommend at least 6 months of expenses)
Insurance (health related, redundancy (although we don't have this as we could survive on one salary only) and death, not too expensive if started young and incredibly important if the worst happened)
Finally, and most important to me, maintaining my own financial independence (actually largely due to a thread I started on here about working or staying at home post DC, wonderful advice revieved). I don't need to work but I do because I need to know I could support myself and my DC if DH left/died.
I appreciate I haven't answered your OP, probably because there isn't a sensible answer other than 'struggle on trying to give DC the best life they can with the benefits available', but I do think there is a lot to be said for protecting yourself from the situations highlighted in your OP.