I think this rather alarmist article on the risk to American and global GDP from declining birth rates and the shift in population mix towards older demographics is a great example of a major trope in the American media. This is specifically in regards to focusing on total GDP growth as the end all goal of the economy rather than the much more relevant to your daily life GDP-per-capita growth.
To really think about total GDP can be so deceptive, imagine that you have an economy with 100 people where everyone makes $50k/year (e.g. a total economy with a GDP of $5MM/year). Imagine that that suddenly, you add 20 new people to your economy and the average income drops to $45k/year. Now your GDP has grown from $5MM before to $5.4MM (an 8% growth rate!), but average incomes have actually fallen by 10%. If all you care about is GDP, then you'd see this as a great win, but the average person is now worse off than before and the economy is moving people in the wrong direction.
Obviously the real world is even more complicated than this simple example. And to further muddy the waters, the actual number we really care about is not per-capita income per se, but rather the very hard to quantify quality of life. For instance, quality of life could even increase even while individual incomes are declining (if for instance cost of goods was falling faster or that income was being earned in a shorter period of time). The rise of the internet for example has brought consumers many benefits (think Facebook, email, YouTube, blogging, etc) that are largely free to consumers, thus boosting the individual's quality of life without any actual additional income needed. So focusing on per-capita GDP has it's shortfalls, however the focus on GDP as a whole I think is missing a very important part of the goal of our economy and gets people worried about some sometimes absurd things (like a fall in the size of the population).
Take for instance the case of Japan and its so called lost decades since the beginning of 1990's. Japan of course has been facing its own demographic changes including a much lower birth rate, very little immigration, and a heavy shift in population towards older age groups. As a result of these changes and the bursting of an asset bubble in the late 1980's, the economy has been stagnant more or less ever since. In fact, in this light Japan is simply several decades ahead of the US with more or less the same forces at work. But when you look at the per-capita GDP's, Japan has actually done rather well over this time and has seen about as much growth as the US since then.
So all of this means that I think our fears the slowing birth rates will lead to slowing growth in individual's quality of life is rather overblown. Furthermore, a shift in population from younger wage earners to older retirees, doesn't necessarily mean quality of life will decline, even if total labor income declines. This is because many in this future cohort of retirees have been saving for decades through social security, retirement accounts, and pensions to ensure they can continue consuming after leaving the labor force. So assuming that a broad swath of the population retires and gradually begins dipping into their saving to pay for consumption, then total consumption will actually fall less than the supply of labor due to this shift. To some extent the economy will certainly attempt to compensate by shifting production to economies with a higher supply of labor (this could be a big win for Africa in the coming decades), but this kind of production has already largely left the US long ago and for the remaining services wages will thus be forced to rise. Thus those who are younger and still in their high production years, they may actually see big increases in total income, likely far exceeding the increase in cost of living.
In addition, retirees on fixed incomes will be seeking to continue maximizing their quality of life at the lowest cost possible, creating strong demand for cheaper alternatives to today's labor heavy service industries. This could be a big win for various automation technologies that are currently not very economical when compared to today's super cheap and heavily supplied labor pool. Think self driving cars, automated espresso stands, and self check out kiosks at retail establishments everywhere. In fact, a temporary drop in the relative size of the labor pool in developed countries could be just what is needed to help shift the economy to invest in a new layer of labor free infrastructure and lay the ground work for the economy to move labor into higher marginal productivity roles in future growth spurts.
The article makes this point in a much more indirect fashion: "Business investment has been lower than usual in recent years, but that may be a consequence of the 'demographic premium' that's about to go into reverse. In recent decades, high support ratios plus rapid globalization created a huge increase in the effective worldwide supply of labor. You'd expect a glut of cheap labor to suppress investment. This is now about to flip. Because of the age transition, the global supply of labor will grow more slowly. This should support the demand for investment."
In the end, changing demographics will certainly represent big opportunities for entrepreneurs in the right place. Many people have focused on the opportunities in health care or retirement communities (which will surely see growth in demand). To me however, the real big winners will be the the people who are positioned to replace high-labor low-capital services with equivalent low-labor high-capital services. They will be able to take advantage of pricing pressure that will be faced by competing firms who stick with labor-intensive practices in an environment of dwindling labor supply and rapidly rising wages.