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.