Economic collapse, to a greater or lesser extant depending on how fast adjustments are made. Though in some cases adjustments cannot be made. Worst case societal collapse (think violent revolution).
Pretty much the entire world economy is based on growth. Individual countries economies for the most part are also based on growth. In either case part of the growth is in population so there are more consumers. Additional most societal institutions and jobs require having a certain number of people to function for everyone. Different countries have different critical jobs and institutions. Care for older population is a big one in most places, doctors, nurses, in home care, and people to do things for the old they can't do anymore. Too few young people means likely too few of those people to take care of older population. That in turn either means the state has to pay more to get more people in those jobs, or care falls upon family which can force them to work less (or quit completely). More money spent by government means less spent somewhere else, some of that will be critical or at least inconvenient for someone. Family working less, or quitting altogether, means they are no longer adding to the economy and become a drag. Further a ballooning older population can lead to a drastic drop in tax revenue and compound the drag on the economy they are already having. GDP can drop which can devalue a currency, then leading to increased costs for imports and borrowing. This can further discourage people who would otherwise have children to not have any. Once this gets into a positive feedback loop it can continue to get worse faster than a society can adjust.
Everything is interconnected in our economy inside any one country, but also across the entire world. A positive feedback loop (like the mortgage crisis the US) can lead to a recession, or worse a depression. Then people are out of work and might not be able to afford the means to continue living, they then can become desperate. This can lead to a crisis and even revolutions (has happened before).
Too big a drop in population guarantied to cause societal collapse? Of course not. It doesn't even guarantee economic collapse, might just be a recession where most people survive fine in the long term. It might all be fine. What the outcome is really depends on how well positive feed back loops caused by a drop in population are handled, and if they happen slow enough they can be handled. Lots of the Western world is in trouble, but a population drop might help climate change, it also might not if a positive feedback loop (permafrost methane) starts accelerating climate change.
I'm not an economist so I will take your words on this, though I still struggle to believe it's an issue and have some remarks.
I am not worried about needing enought people for jobs. Given advances in inteligence automation then we can't forever have enough jobs for all humans. If a country can impliment a universal basic income then the citizens can at least have a basic living.
Why are most countries based on growth? That appears reckless. Unless we expand into space then at least population growth caps out at some point. Doesn't every other growth have a limit?
Unfortunately, I think that capitalism is here to stay, so things will just get shittier and shittier for everyone. As others responding to the top level comment have mentioned, declining birth rates means more stress on the entire system, where we'll see more young people without any future to hope for since all their energy and money will be coopted for caring for old people, old people having shittier end-of-life experiences because there isn't enough money to support them, and countries will not able to support anyone because there's no investment due to lack of growth.
Your remarks are spot on. They are why I've read up on some of these problems over the years, even though I'm not an economist.
Automation very well might mitigate and/or cause other issues. It is to be seen if a capitalistic system will succeed in being reasonable, especially some of the more virulently capitalistic ones like the US. People being more productive has avoided many problems in capitalism for a long time, AI is a new way for this to happen.
Universal income is an excellent idea. There have been some really convincing studies where it has been implemented on small scales (one town or village). So far it hasn't gone much farther as there are strong contingents of people unreasonably against the idea.
Basing economy on growth is problematic. Growth being key to capitalism has been a criticism for awhile. It is reckless, doesn't reflect actual reality of resource limits of growth, and sets up problems some countries are facing (declining birthrate, job displacement due to automation, etc).
The fundamental issue with declining populations - fundamental as in regardless of the economic system of the country - is decreasing standard of living.
The very simple metric is productivity-adjusted hours worked per person. This invariably falls in cases where overall population is declining, because populations age as they decline, and older people work less (retirement) than younger ones.
As this metric falls, the country's economy basically just produces less stuff per-person than it did in the past. This makes everyone effectively poorer.
In extreme cases, there can also be issues with availability of services. E.g. healthcare: Each doctor/nurse/caregiver can only effectively attend to so many patients and this number is difficult to increase with technology.
Standard of living is supported by those who can produce versus those who cannot. As population declines the demographics skew to mostly be older non-working people. There is a certain point where the percentage of people working versus not working is too small, then the economy can no longer produce enough for everyone's current standard of living. It can range from relatively minor case of not being able to get all the variety of food, or it can be major where people starve because not enough food can be produced. Or medicine, or care, or electricity, or oil, or plastic, or TV shows, etc.
Given enough time a new equilibrium and standard of living comparable to the old one will likely result, but getting to that new standard of living can mean people died.
Would it have to be extreme where people are straving? A nation already wealthy has a lot of infrastructure which just needs to be maintained or adapted?