This question is so tricky, and there is not one universal answer. Relating to Harrington's Rule, anytime stored seeds are exposed to higher temperatures, they will begin aging faster. When the temperature goes back down, they will return to aging very slowly. So if the power is down for only a couple hours – and the freezers, refrigerators, and seed bank doors are all kept closed to keep the temperature lower inside – the damage may be minimal for most species.
If a restoration seed bank’s power is down for many hours or days, the damage may be minimal for some species and severe for other species. Because of the seed survival curve (sigmoid curve = a gradual decline followed by a steep drop in viability) and the fact that a seed bank contains seeds stored for different amounts of time, certain seed accessions may be exposed to that steep drop by a power outage that speeds aging. Therefore, this consideration is not just species-specific, but accession-specific too.
In any power outage, make sure to keep doors closed on refrigerators, freezers, and the seed bank itself to keep the temperature and relative humidity as close to their targets as possible. Aside from restoring power quickly, this is the best thing you can do to minimize damage to your seeds.
If you work with species where research has given us some idea of how long the seeds will survive in storage, we might be able to advise you on what length of power outage means you need to prioritize testing/using the stored seeds (especially older accessions). But more importantly, we need to make power outages very short and very rare events! This is why a backup generator is an essential piece of equipment for all seed banks, and an automatic-start backup generator is ideal.