As this question seems to be more a question of social views rather than technical aspects, I'll avoid the latter for this question.
To my knowledge, aspergers is not really the preferred term these days, and has been phased out in favor of just referring to it as Autism Spectrum Disorder (ASD), and so will refer to it thusly.
So, to start, designer babies are an interesting thing. Although we've had the power to do gene editing for decades now, experimentation on humans has been very slow and lethargic.
People get rather apprehensive about it, and rightly so. At minimum it is easy to mess something up accidentally, and give someone cancer. If you take it to a different area of concern, there are many who feel that it is an act of playing God, that it is arrogantly saying we can "fix" what he created.
A whole breadth of these kinds of moral concerns will come from everyone. In the end though, designer babies will become a thing - as there is the moral problem on the other side, of if you have the power to do something like remove the genetic inclination for a family history of heart disease in your child, and you are not, then... that's rather disconcerting...
And so, designer babies will likely start off here. Let's cure cystic fibrosis, down syndrome, and other things we know are not desired.
But ASD can get really complicated, especially when you get onto the tail end of things. I have some sort of ASD disorder, likely executive functioning issues. I hate that it makes it harder for me to be motivated to do anything, and makes planning difficult also.
And yet I also possess a strong ability to remember tons of facts that I can draw from at any time. I have a pretty good ability to pick up a lot of different creative interests.
My point is, at what point does it stop being a "disorder" and starts becoming "personality?"
These kind of questions are what scientists, doctors, and parents will have to reason with. Are the beneficial sides of ASD separate from the negatives, or are they connected?
If I give my child ASD to some degree on purpose, and then he or she grows up to be extremely antisocial and unhappy, are you the parent to blame for their problems?
These kind of questions are going to give a lot of parents pause. There will be some who are conspiratorial, refusing to "jab" their children for anything, even cancer. There will be others who are super impulsive "I can make my child's skin blue! That sounds awesome!"
I think most people will allow some genetic therapy when it becomes available, especially if the disorder is something severe like cancer, cystic fibrosis, crohn's disease, etc.
I can imagine there being a big push against "aesthetic" or unnecessary genetic modifications. The idea of that sounds extremely dystopian to a lot of people, and so I imagine a number of people will resist it.
I can see it leading to increased interest in post birth modification, allowing consenting adults to do whatever they want with themselves, maybe using something like an engineered virus as a delivery system.
This of course, is all speculation. People will do strange things, and this century will have a number of great transformations in genetics.
I think that the pandemic however has been a good microcosm of what to look for. Not necessarily people reacting in the exact same way, but showing what kinds of people will likely do what.