In short: No.
The idea that the phases of the moon affect the psyche of human beings - which though having existed since ancient times - has never been proven to be accurate. In fact in studies dealing with, amongst other things, schizophrenic episodes and epileptic fits, no correlation was found between the worsening/increase of the latter and the full moon.
The few things where a (usually very small) correlation was found (for example poorer quality of sleep during nights with a the full moon) can easily be explained as being a result of a darker/brighter night size (for instance it's obviously harder to have a good night sleep when the night sky is being lit up by a full moon). In addition in the case of the various infamous "the full moon causes crime" studies, poor methodology has consistently been shown to be what is at play.
So essentially, especially since we already tend to use artificial light sources to illuminate our cities, there would be no significant direct affects, due to a darkened moon on human activities.
There are however some species which change their behaviour slightly depending on the phase of the moon.
For instance, predators which usually hunt during the night are more likely to hunt during the day on days before and after a full moon. This again is likely nothing mystical, it's thought to be because hunting on full moon nights is harder than on very dark nights (as it's easier for prey to spot them when its brighter). To compensate for this, predators sometimes also hunt during the days before and after the full moon.
Many species also prefer mating on nights which don't have a full moon, mating often makes creatures more vulnerable to predators and so it's better to do it when visibility is low.
Now you may have noticed that both of detail changes in behavior due to nocturnal creatures being negatively affected by increased illumination. So in these cases never having a full moon would actually be a good thing.
Now all that said, there is a particular biome where this darkened moon might have a significant influence: The Great Barrier reef off the coast of australia. When the corals start their annual en-masse spawnings (when they release their sex cells into the water in order to reproduce) depends in part on the full moon. Corals nearer to the coast spawn during the week after the full moon in October, and the corals further out to sea spawn after a full moon in November or December. These events are in part also directed by other factors such as ocean currents, temperature and salinity but the full moon remains one of the most promenant and unchanging. Because we aren't all too sure on how the corals synchronize their reproductions so well we also don't know how darkening the moon would perturb this concurrence. It might only be minor or it could be increadibly damaging.
Regardless, The Great Barrier Reef (a varitable bastion of incredible diversity) is already under serious threat due to climate change and introducing yet another discordant factor could be truly devastating. If your future people/governments are ecologically-minded, conserving this ecosystem (it it still exists by then) in the face of the darkened moon should be one of their main concerns.
So in conclusion: while this dark moon would have barely any direct effects on humans, it could still slightly alter the subtle going-abouts of some creatures (the lack of a full moon would be mostly a positive thing for the animals concerned however). The only, potentially major effect would be on the well-being of one of the largest marine ecosystems: The Great Barrier Reef.