That seems to be the case in an article on the recent drop in fertility rates that warns:
"If the trend (lower fertility) continues — and experts disagree on whether it will — the country could face economic and cultural turmoil."
That is more than a bit hard to see. If we do see a sustained drop in the fertility rate it will mean that eventually we will have higher rates of retirees to workers, assume no offsetting increase in immigration or an increase in labor force participation by either the prime-age population (ages 25 to 54) or older potential workers.
However, the economic implications of this rise in the ratio of retirees to workers are very modest. According to the Social Security Trustees Report, the impact of a sustained fall in the fertility rate would increase Social Security's projected shortfall over the next 75 years by an amount equal to 0.36 percent of payroll over this period. This is equal roughly 0.12 percent of projected GDP. There are other costs, such as Medicare, that would also increase with a larger ratio of retirees to workers; however, this would be offset in part by reduced spending on education and health care for the young.
By comparison, the increase in military spending associated with the wars in Iraq and Afghanistan was close to 2.0 percentage points of GDP. While these wars have prompted some opposition and protest, it has not led to economic turmoil. It is difficult to see why an increase in spending that is perhaps one-tenth as large would be expected to cause economic turmoil.
It is also worth noting that plausible changes in productivity growth swamp the impact of even large changes in fertility rates. If the country, had sustained the rate of productivity growth it experienced from 1995 to 2005 (also from 1947 to 1973) over the last twelve years, it would have the equivalent effect on workers' take-home pay as reducing the Social Security tax by 10 percentage points. If the rate of productivity can be boosted by just 0.1 percentage point, it would swamp the long-term impact of a lower fertility rate on workers' living standards. And, this is before even taking into the account the benefits of reduced stress on infrastructure and the environment.
In short, we should worry if people don't have children because they don't think they can afford them. We need not worry about running out of people.