The following is a table supplied to be by MSD under the OIA.
See also MSD publication Children in Families Supported by Main Benefits:
The proportion included in a benefit at birth or very soon after fell from around 25% of children born in the 1990s to 20% of children born in 2005 and 2006 and 18% of children born in 2007. Although contact with the benefit system fell, as many as one in five children turning 15 in 2008 are estimated to have been supported by a main benefit for a total of seven or more of their first 14 years of life. An estimated one in ten spent a total of 11 or more of their first 14 years supported by a main benefit.
So in the year when NZ had the lowest unemployment rate in the OECD, 18 percent of children born that year were dependent on welfare by December 31.
And regarding increasing benefit levels being a disincentive to work:
“…every 1 percentage point increase in the level of poverty reduction achieved by the welfare state is associated with an increase in the number of jobless families by 0.63 percentage points. Among the English-speaking countries, the correlation is even stronger (about 0.92), so that Australia and the United Kingdom reduce child poverty very significantly and have very high levels of joblessness among families; while Canada and the United States reduce poverty much less, but have much lower levels of joblessness.”
Whiteford, P. and W.Adema (2007), “What Works Best in Reducing Child Poverty: A Benefit or Work Strategy?”, OECD Social, Employment and Migration Working Papers, No.51, OECD Publishing. http://dx.doi.org/10.1787/233310267230
Table not enough. Here is the full OIA as requested.