There have been two big blips in panic about population. In the Early 1800s, Maltus looked at a chart of growth of population which was curved because it was cubic against a graph of growth which was linear. There is a point where such a graph intersects and population grows beyond the resources we can get at easily. Malthus (in his first edition, toned down in subsequent editions) that aid to the poor was immoral because it pushed the world towards that point. The industrial revolution saved that from happening, as mechanization made resource use more efficient.
Then in about the 1960s, there was further panic, basically looking at the same kind of graph. This one was based on the idea that food production globally was peaking, and population growth was still going on. Thinkers like Paul Ehrlich and consortia like The Club of Rome put out books with titles like "The Population Bomb" arguing that famine was neigh. Of course that didn't come to pass because of the Green Revolution.
The problem is that we can't count on technology to save us all the time. The current "revolution" in information is more about contentedness and less about access to resources (Though there are things that will be able to help the so called "bottom billion" since they're connected through mobile phones and missed the desktop era). The one thing that saves us is that as countries develop, birth rates go down. So population growth slows. As more countries develop, the global growth slows. This is good in terms of resource depletion. It is also problematic because capitalism as we know it depends on growth. There are two parts of growth in a capitalist economy. The first is technological growth, something called by economist as Total Factor Productivity. We get better at making things. The other is population growth. So if a country isn't growing, it feels like a recession. Japan has had this problem for decades, and now it looks like the west is facing the same problem. One solution is immigration, but a lot of immigration dilutes a homogeneous culture, and you get lots of backlash over it.
Ultimately, I'm not sure at what level we can say that population growth and levels are a problem. Especially because if you look at potential solutions, there's nothing good. There is a lot of push back against the One Child policy in China, and it has shaped demographic trends that will echo for decades. New entrants into the Chinese workforce are due to peak pretty soon, and now they have a bunch of old people to support. It's not a pretty picture. If it is a problem, it would take a large coordinated effort to solve. And that assuming the climate at stasis.
What's probably going to happen is a crisis point pushed by climate change that kills a mass of the poorest people on earth, and that might get the ball rolling in the rich countries.