Reputation is fantastic at solving collective action problems. Climate change is the biggest collective action problem in our species' history. So what can reputation do to address climate change? At an individual level, perhaps a bit - there seem to be reputational rewards from shopping green, and from climate-conscious generosity. That said, conspicuous consumption is always likely to be a more reliable status symbol than conspicuous non-consumption.
But me not taking leisure flights (which I haven't for a few years) is not going to stop China building coal-fired power stations. So how do we get international agreement to limit emissions? The world of international relations has been described as one of anarchy and self-help. And yet, most countries honor most of their treaties, most of the time. In the absence of the world government that isn't going to happen any time soon, concern for their reputations is one of the things that makes them do so. (I am a fan of UC Berkeley law professor Andrew Guzman's papers on this topic.)
Unfortunately, Guzman (I think it was him) has written that reputation seems to work best at encouraging compliance in areas that aren't central to a country's interests, such as nature conservation. I don't think we can really put the transition to a low-carbon economy in that category. The higher the stakes, the greater the temptation to break the rules (I think the Dutch football team's behaviour in the last world cup final is a great example of this.)
And it works best when there's also domestic pressure on the government to behave well. Which is missing at the moment. And unlike people, countries rarely treat each other in one area based on how they behaved in another - rather, they have lots of different reputations.
So in terms of avoiding a global tragedy of the commons, most of what's in our social nature is still pointing the wrong way. I'm still hoping for some moral leadership to come from somewhere. But I'm not hopeful.