Fantastic question, and one that is extremely pertinent today (and frequently discussed as a result!) because so many children are actually working, right now, in the third world.
Economic historians are rightly obsessed with this question, as are many other groups. To quote some passages, with the reference articles attached that are pertinent to your question (and subsequent answer):
We rely on economic theory to assess whether statistically significant changes stem from minimum age regulation. In the presence of competitive labor markets and working capital within the household, we show that binding minimum age of employment regulations shift child work to the family farm or business but have no effect on schooling. Our argument is based on three assumptions. First, child and adult labor are perfect substitutes subject to a productivity shifter (“the substitution axiom” in Basu and Van 1998). Second, the child’s household can freely substitute adult and child labor between productive tasks inside the household, i.e. “non-saturation.” Third, adult labor can move freely between the household and the labor market, i.e. “competitive adult labor markets”.
The Impact of Minimum Age of Employment Regulation on Child Labor and Schooling, Edmonds and Shrestha, here.
On an argument against legislative solutions on child labor. While its about the British evolution, it serves as a useful light on what could have also happened in the U.S.
Third, the historical evidence of industrial revolution in Britain proves that legislation does not really have an impact on child labor. Deterioration of child labor came before the introduction of such legislation. This was again caused mainly by economic factors as increasing sophistication of technology, higher productivity and higher wages that reduced the need to send children to work. In addition, Nardinelli argues that decisive factor response for reducing the supply of child labour was the rise of the living standards associated with economic progress, which came first.
Child Lbaour in Economic Perspective, Trnik, here.
Another thing to realize is that child labor in the United States has always been a multi-faceted process.
In Hindman's Child Labor: An American History, he argues that prior to the Industrial Revolution, child labor was not just a supplement to familial income, but also served as a solution that covered many bases. Agriculture as a whole was very suitable for education: you have circular periods with predictable downtime, so it was easier to know exactly when children would be most receptive to education (as in having the most time; c.f. p.14). Indentured servitude wasn't just a means of providing families with additional income – it was an effective strategy against the problem of orphans and children of poor and dependent parents in England and continental Europe that soon had quite the following in the U.S. (c.f. p. 15-16), and later, the Industrial Revolution provided a similar solution for the poor and displaced women and children particularly.
Also important to note is that legislative action itself wasn't just a one step process, but rather a path that first focused on the regulation of actual labor conditions, hours, and what can and cannot be delegated rather than dealing with the regulation of an entire group as a whole, which came later.
So really, when you take all these factors into play, the argument comes out a bit mixed. Yes, it did have somewhat of an effect on U.S. family incomes. But how much? We really can't say. Various regions in the U.S. were so remarkedly different in their usage of children participating in the workforce that its hard to even draw a united conclusion, much less a satisfactory one. Even today, there are still a lot of arguments being made for or against one case, and the conclusions are still very much up for debate because so many factors were all playing out at the same time (technological development, wage increases (minimum wage laws were being passed at the exact same time), etc.)
But you should read up on this too! Here are the sources, as well as some additional ones:
http://www.unicef-irc.org/publications/pdf/hisper_childlabour.pdf
http://users.nber.org/~rdehejia/papers/dehejia_gatti.pdf (More of a modern day perspective, but still useful)
http://www.ilo.org/wcmsp5/groups/public/---dgreports/---inst/documents/publication/wcms_193680.pdf
http://www.bls.gov/mlr/1990/03/art5full.pdf
http://www.jstor.org/discover/10.2307/2564873?uid=3739256&uid=2&uid=4&sid=21103851913201
http://federation.ens.fr/ydepot/semin/texte1112/JAN2012CHI.pdf
http://www.fordschool.umich.edu/rsie/workingpapers/Papers476-500/r486.pdf
http://old.econ.ucdavis.edu/faculty/gclark/210a/readings/clark-ysbrand.pdf
Child Labor in America: History, Policy and Legislative Issues By William G. Whittaker