I'll give you my perspective. Developers should care about docker as there are other developers who are willing to use docker and have already built an expertise in it. They are willing to take up the roles of a DevOps engineer along with being a developer. So the Ops part of DevOps is what they are now building expertise on.
These days, you'll find more and more guys who can develop, orchestrate, automate tests, automate jobs and build tools to monitor and take this complete package into production single-handedly. These are the guys who are pushing docker and other tools among the developer community.
Also, the tide of market is towards virtualization, auto-scaling, automation, machine learning and docker fits in all of these. It has become very imperative to use docker. The businesses are willing to pay 2x for a single guy who takes all these responsibilities and when there is demand for such guys, the supply will also begin. This is from point of view of an employee-employer.
Technically, in the organisations I have worked, there are separate development and DevOps teams, although they work very closely for deliveries.
The DevOps engineers and developers share a vast majority of skill-sets here and hence there is a negotiation of duties sometimes.
The bare minimum a developer can do is to share his binaries, but he needs to understand that the binaries will be used to run inside a docker container and for that he needs to understand how docker works. For kubes, swarms, mesos etc, the developer may not even care what's being used, but basics of docker should be very well understood by the developer and a mindset should be there from the beginning to build the application loosely coupled for re-use as micro-services. If the application is built from that mindset (which requires basics of docker), then the DevOps engineers can take it up from there to auto-scale, orchestrate, test, deploy and monitor.
Also, most of the times there is no one size fits all kind of thing. A developer does not know clearly how to build a docker friendly app and a DevOps engineer quite rightly does not know the internals of the app building process. Hence, most of the times, organisations prefer to give both these tasks to the same guy to speed things up. If there are separate things, then a continuous feedback mechanism is required from the DevOps team to the dev team for making the apps more futuristic and docker/cloud/scaling ready.