I would argue that the best way to deal with population growth in the world today is to promote economic growth. There is nothing wrong with, at the same time, trying to promote things like contraception. However, economic growth is the best “cure” for population growth. This is true for at least two reasons.
First, it is true because population growth is largely an economic problem (though too much growth may end up being an environmental problem). The problem is not so much that there are too many people as it is that there are too many poor people. If we could successfully promote economic growth, the larger populations would be able to enjoy better standards of living and a major part of the problem would be solved.
Second, economic growth is a way to cut down on population growth. In essentially all societies from which we have evidence, people have fewer children on average as they grow richer. We are not completely sure why this is, but there is definitely a connection between countries getting richer and their birth rates falling.
Of course, it is not easy to cause economic growth. However, it is equally difficult to change people’s cultures and encourage them to have fewer children. It seems to me that economic growth is a more feasible strategy.