The meaning of "apace" as it is used in the opening paragraph of the passage found in the article of "the cost of children" by Ruth Mace

It means swiftly, rapidly without stopping. Mace wants to say that the decline of birth rates is a modern trend that grows rapidly all across the world, along with the rise of education, industrialization, urbanization, medicine. All of those things are making our lives easier. Still, we opt for fewer children and smaller families.

Apace in the context of the article means that there is a continuous, steady and quick change seen in the world when it comes to the decline in birth rate and family size which has been prevalent or common social change for the past two centuries.