Not really - or rather, I'm not quite sure what would consitute "evidence" for moral questions.
To take a simpler example, most people would agree that it's the right thing to jump into a river to save a drowning toddler, even if there's a slight risk to your own life (the current is strong) and it's certain to damage your clothes and smartphone. Yet I don't see what would consitute "evidence" that it's the right thing to do.
(edit) But if you're asking about reasons for believing it's the right thing, I'd say that the future world would likely be a better place if wealthy, educated and responsible people have more kids.
So I found this post quite interesting:
http://www.gnxp.com/blog/2009/03/gnxp-readers-do-not-breed.php
(I'm quite sure that the demographics of this site closely parallel the demographics on Gene Expression).
Research seems to indicate that people are happiest when they're married, but that each child imposes a net decrease in happiness (parents in fact, enjoy a boost in happiness once their children leave the house). It's possible, of course, that adult children may be pleasurable to interact with, but it seems that in many cases, the parents want to interact with the children more than the children want to interact with the parent (although daughters generally seem more interactive with their parents).
So how do you think being child-free relates to rationality/happiness? Of course, Bryan Caplan (who is pro-natalist) cites research (from Judith Rich Harris) saying that parents really have less influence over their children than they think they have (so it's a good idea for parents to spend less effort in trying to "mold" their children, since their efforts will inevitably result in much frustration). And in fact, if parents did this, it's possible that they may beat the average.
(This doesn't convince me in my specific case, however, and I'm still committed to not having children).