Maybe the fish should be the property of society and the location of the fish is the catchers right, with safe handling and responsible returns being their responsibility?
Step one and two describe how Society currently works in civilized countries. The fishing territorial Waters belong to the nation, and fisherman are licensed to use it.
I don't know what you mean by responsibility here. In the current system fisherman are incentivized to catch fish by the profit* from selling them, which they pay taxes on. You need incentives because picking people and forcing them to be responsible for fishing would be slavery
