I'd think a drum brake would be the most durable braking system; three pairs of shoes will last a lifetime and the drum itself will likely never need replacing.
Scratch that, a coaster brake will last even longer than drum - the friction material is brass! Should last a lifetime and are mechanically even simpler and more robust than drums.
Scratch that, a coaster brake will last even longer than drum - the friction material is brass! Should last a lifetime and are mechanically even simpler and more robust than drums.