To figure out how to get humans to voluntarily stop, you first need to know why humans reproduce. Some of the reasons off the top of my head:
- Sex is fun (= body produces drugging hormone cocktail to ensure humans do it as often as possible to make the race survive)
- Babies are cute and lovable and yes, there are people who love children and love working with children (= body produces yet more hormones to ensure humans take care of noisy and aggravating bundles of helplessness in order for them to grow up to become proper, sexually active humans)
- Security when getting old (= who will be my doctor when I am old and fragile, who will produce food when everyone is getting old and infirm, who will perform all the services that keep society alive when everyone is over 70 years of age?)
- leave an impact on the world for posterity (= make sure one's genes are spread as wide and far as possible)
Sex is fun
Give people either a substitute or a 100% effective way of birth control. A substitute can be a drug that gives a much better high than sex, or some kind of advanced sex bot that is just so much better at it than normal humans that people choose the bot every time.
Babies are cute
Try and find a substitute - a baby bot that smells right, feels right, acts right, grows right and is a lot less noisy and aggravating than a real baby. On the other hand, it might be a bit hard to convince someone to put in the necessary work (physical + emotional) for an artificial baby.
Either make everyone immortal or have machines that are high tech enough to slowly replace the workforce when it ages out. And with that I don't only mean machines that can drive a bus, but also machines who can build and repair machines that can drive a bus!
Either true bodily immortality, or provide some kind of personality safe where you digitize people so that they can still influence things 200 years down the line
Solution: The Voluntary Matrix
I think it is very unlikely for all of humanity to accept being replaced by robots. Also, if robots look and feel human enough to be perfect substitutes for sex, children, conversation partners, and work force, it is debatable if those robots aren't human as well.
What might get you better results is creating a total Virtual Reality immersion that is addictive enough that people prefer staying in there to livinig real life. Kind of like "The Sims" on steroids in order to fool our base mammal instincts. Add some gamification for more addictive properties, other virtual worlds, etc.. Make sure that jobs and social interaction only take place in VR (for easier communication no matter where in the world you are, or some other such crap). Make sure that people never actually have to leave VR (no going to the bathroom, or having to do exercise to prevent physical problems), and that they neither want to. Getting old will not be a problem anymore because the VR avatar can be at any age and is not impeded by bodily concerns. The Heritage fact might be satisfied by programming VR stuff that will survive them. Or by giving immortality.
If everyone is in VR 24/7, nobody will have physical children anymore.