Look at this graph:
Remember how I was talking about how awesome it was that we could eradicate diseases, but there are some potential downsides? That right there is the potential downside. Since we started getting good at fighting disease and starvation, the world human population has uncontrollably skyrocketed. We see this kind of population growth in nature all the time when a species is introduced to an environment that has easy access to food and no natural predators. When it happens to other animals, though, they eventually burn through all their resources and start starving to death, causing a decline in population almost as precipitous as the rise, sometimes even eradicating the population altogether. That hasn't happened to us. Not yet, anyway.
We've avoided the Big Nasty Event so far because we, as a species, are really goddamn clever. We have an answer for everything nature can throw at us. We meet diseases with medicine, famine with genetically modified crops, drought with irrigation, and predators with guns. We've prevented so much death. It's amazing, and it's wonderful, but in the long run it may also be dangerous. It's not hard to see why.
Every time we remove a threat, we remove an obstacle to population growth. It's a big, fancy feedback loop. Eventually, the cycle will break. Something is going to come up that we don't have an answer for, and it's going to wreck our shit in a very alarming way. So how do we actually fix that? The short answer is, we probably don't. We could use population-controlling methods like China, but that probably won't fly in a democracy, and it would do some pretty serious harm to the economy. Sure, the economy is less important than human continuation, but it's also a more palpable and immediate concern, so good luck convincing people that they need to give up their livelihoods for the greater good. I wouldn't, and I'm the one sitting here ranting about how important it is.
The solution I like throw around when I talk about this is colonizing other planets. The most feasible solution I can think of is, "I dunno. Space?" And even that just buys more time.
I mean, seriously, look at that graph!