Different people have different reasons
for valuing a stable climate.
Economic Security
The World Bank states, "Annual total damages from disasters have been increasing for decades and models show that population growth and rapid urbanization could put 1.3 billion people and $158 trillion in assets at risk from river and coastal floods by 2050."
Unmitigated climate change is expected to lower world GDP by 23% for the year 2100. In today's terms, that would represent about $17 trillion each year in lost global revenues.
National Security
The United States military has identified climate change as a "threat multiplier," as increasing extreme weather is already exacerbating droughts and floods and will continue to get worse. Climate change is expected to make Middle East droughts more severe and longer-lasting, and droughts have already exacerbated economic and social unrest that contributed to wars in Syria, Sudan/South Sudan, Somalia, and Darfur.
Border Security
As noted in the section on national security, extreme weather events related to climate change have already been linked to wars.
Wars, of course, result in large numbers of refugees. The UN states, "65.3 million individuals were forcibly displaced worldwide as a result of persecution, conflict, generalized violence, or human rights violations."
This destabilized entire regions, even as far as the European Union, where Hungary has closed its borders, Brexit passed for the purpose of closing Britain's border, and even Germany has essentially closed its southern border to refugees. This is impinging on a core value of the European Union, open borders.
Ecological
On land, the increasing warmth, changing drought and flooding patterns, and poleward movement of habitat places substantial stress on species, especially considering that roads, cities, agriculture, and more make it difficult for many species to migrate.
In the ocean, climate change is radically altering ecological systems via rising temperatures, increasing acidity, decreasing oxygen content, collapse of coral, and decline in phytoplankton (the very base of the food chain) and krill (the crucial link between microscopic food source and macroscopic creatures). In fact, scientists have found that "today's oceans are absorbing carbon about an order of magnitude faster than the worst case in the geologic record -- the end-Permian extinction," when 96% of ocean species perished in the worst extinction event in planetary history.
All this and more adds up to threaten a human-caused - and optional - extinction cascade.
Winter Sports
Alaska has lost 4 to 6 weeks of its winter season. People engaging in winter sports now travel farther, or they simply participate less in the sports they love. Companies meeting their sports needs are also feeling the pinch of winter's decline.
The group "Protect Our Winters" includes many professional and Olympic athletes whose livelihoods and lifestyles are being eroded by climate change. For one example, US gold medalist Jessie Diggins discusses the dramatic changes she has witnessed: "Skiing was a way of life there and I fit right in. But by the time I graduated high school, things were starting to change. None of our regular ski trails had enough snow to last the season. Racing and training started to mean long days in the car just to ski on a two-kilometer loop of man-made snow. We spent more time driving to ski than we actually got to spend skiing...[Now] you can’t even think about hosting a World Cup race without having snow-making capabilities. Think about that for a second. You can’t host a winter sporting event…without fake snow. "
Care for Others
Society's most vulnerable people will be hurt most by climate change. It is easier for wealthy people to add air conditioning, to move out of a flooding area, or even to relocate to another country. Additionally, increasing climate instability hits poor farmers hardest, climate refugees are more likely to be poor, and poorer areas of the world are expected to lose most of their anticipated growth in GDP.