It's undeniably a lot more expensive than it used to be. I don't think it is the main reason that people are having less children but I think it may be a big factor in why certain expensive cities are seeing big declines in young families. Places like Brighton and York have seen large declines in the number of children from the 2011 census to the one released this summer.
France was arguably the first country to have a fertility rate decline in the 19th century. The main reason seems to be the rise of secularism.