How do we prevent the extinction of Europeans?
As far as I know, Europeans are in no danger of "going extinct", so no specific actions are required to accomplish this.

The species Homo Sapiens as a whole is at very low risk of extinction compared to thousands if not millions of other species.
That said, the biggest risks Homo Sapiens face today are likely human-made climate change, irresponsible and wasteful use of resources at rates greater than they can be recovered, and a threat of weapons of mass destruction and thermonuclear war.
To some extent, these issues are interrelated, as global societal pressures and distortions which might lead to armed conflicts and wars are increasingly fueled by local and global crises that themselves are symptoms of climate change caused by shortsighted or irresponsible practices.

In order to make it possible for humanity to not merely survive but also thrive in the long term, It is advisable to focus on big systemic changes to avoid the worst outcomes of climate change and consequent societal change, as well as on livability:

- switch to carbon neutral production chains, as well as ones involving less toxic waste
- avoid fossil fuels for heating, electricity generation, or transportation
- switch to renewable energy sources
- make these energy sources affordable even for countries that struggle with high rates of poverty
- retool electric grids to better handle these renewable energy sources
- increase the efficiency and longevity of devices
- avoid wasteful materials such as single use plastic
- reduce, reuse, recycle (in that order of importance)
- avoid destruction of ecosystems that are known carbon sinks or help protect shorelines, regulate the water cycle or provide clean water
- reconstruct damaged ecosystems
- generally protect and foster biodiversity
- design places where people are supposed to live to be actually affordably livable
- build strong, resilient communities that are respectful of of the local norms and culture
- reform infrastructure to center less on cars and more on people, focusing on public transport options, bikes, and pedestrian safety and comfort
- minimize food waste
- switch to food sources that are lower in carbon production and water need
- improve social security and health care
- normalize and invest in mental health care in particular
- invest in preventative, proactive health care in particular
- recognize things like parks and local nature as an important public good and a valuable kind of infrastructure

And many other related interventions. If implemented globally, these sorts of actions would improve living standards, reduce conflicts, and maximize the long term survivability of Homo Sapiens.