I don't think this is true at all.
Birthrates are already declining. All you have to do is give proper sex ed and easy access to birth control and populations will shrink on their own. You don't even have to begin propaganda around overpopulation, though we may need to tone down the "WE NEED MORE BABIES" propaganda that the right is currently projecting.
The fact is, there are a lot of people (18-29% of non-parent adults in the USA, depending on your source) that don't want children. Give them the tools to make sure accidents don't happen (IMO, vasectomies would be more popular if there weren't so many myths surrounding them), and birthrates will decline naturally.