This question is very broad. If a person is severely depressed and considering children at that moment that does not seem wise personally. Especially if having a child is viewed as panacea out of depression. Regardless of my opinion though I would have no right to interfere and nor should the government. Nor would I shame them for their decision like they are the most vile immoral being on the planet. Besides, shaming people out of presumed immoral behaviour does not work and is more likely to backfire.
I also don't hold the view that because you have any sort of genetic defect you should use that to inform policy and deny the right to have children because of the potential risk. I also don't view depression as solely a genetic or chemical thing either in the first place. So I am even less inclined to want to refuse a person their right to choose on that basis. I am pretty averse to the government being given the power to continue to erode my choices on the pretext of a greater good especially on something as fundamental and part of core biology as having children. What criteria would they use? Would wealth be a measure? Would disability be a measure? Do you trust financial incentive to not skew things? Can you appeal? Do you even trust your government? I sure as fuck don't, no even to fill potholes effectively.
This type of thinking has resulted in some pretty awful things in the interests of a 'greater good.' For the most part, resulting in human rights violations and other unintended deeply alarming consequences. I am not just talking about Nazi eugenics either but much more recent attempts to control population. Like India's state of emergency and China's one-child policy.
There are better methods of reducing birth rates. That don't involve your government potentially violating your human rights, kicking your door in and forcing unwanted medical procedures on you, disregarding your personal choices for some apparent greater good.
Better methods consist of education and lifting people out of poverty. The correlation between reduced poverty and lower fertility rates is well established. You can examine in detail why that turns out to be the case here.
How does the number of children vary across the world and over time? What is driving the rapid global change?
ourworldindata.org
If people want to perpetuate the species that is none of my business at the end of the day. How depressed they are is equally irrelevant. The closest I come to antinatalism was choosing to not roll the dice on producing an offspring that may well have to suffer through bipolar like me. With flawed mental health systems and a judgmental society. This was a devastating conversation to have with my fiance at the time. It is also pretty grim to watch children being vilified for leaving school to protest climate change. What a sad world they are set to inherit at this idiotic rate. Infinite growth on a finite planet can only lead to self cannibalization at some point.