I didn't hear his particular argument, but generally antinatalist arguments go something like this:
1. Suffering is bad
2. Suffering experienced by humans outweighs pleasure
3. Creating new humans is bad
Although myself I'm not planning to have any kids, I don't embrace antinatalism for a few reasons.
Firstly, I don't think most people feel that their suffering outweighs their pleasures. Some do, no doubt, but they're a minority.
Secondly, I don't want to eliminate suffering - at least not at this cost. I would prefer a universe with somewhat suffering humans to a universe without them. I value many aspects of mankind, and want them to persist. More than other species, humans have the capacity to reason, to imagine, to seek knowledge, to appreciate the scale and complexity of the universe, to create awesome machines and works of art. The suffering they experience doesn't outweigh all this for me.
Hmm...
I think the whole anti-natalist argument stemming from the preponderance of human suffering actually leads to a different and more moderate conclusion than the one they posit. Regardless of whether one feels that the balance falls in favor of suffering or pleasure, it's certainly incontrovertible that this balance is still unsatisfactory and can be improved significantly. Anti-natalists have fallen on, imo the very powerful and correct mechanism to most significantly alter this balance. However, I think that removing a few humans from the genepool will have a relatively negligible impact on this balance in comparison to increasing the proportion of responsibly initiated parenthood among the human population. To this end the proliferation of birth control methods including condoms and the availability of reproductive (not to mention educational and other potentially affecting) rights to women might be considered the most important issues to tackle in the world today.