People who are freaking out about the rise in the dependency ratio will normally point to the fact that the ratio of retirees to the working-age population is going to sharply rise. The following graph, taken from Social Security Administration projections, makes it easy to side with the pessimists.
As you can see, by this measure, the dependency ratio has been getting worse for decades, and will approach .4 by 2030.

However, the picture looks very different if you compute the more economically meaningful worker-to-population ratio, as Sheiner, Sichel, and Slifman did in the graph below.
By this measure, dependency has been declining for decades. Yes, dependency is going to start increasing, but it will start increasing from an all-time low. In 2030, we’ll effectively be back at the dependency level we had around 1965.

Unless something changes, the second graph shows that dependency will continue to rise throughout the 21st century. But by historical standards, it should be quite manageable even in 2080.