What is the PSU?
If it's a dumb linear PSU (i.e., iron transformer, diodes and cap), or battery, it is capable of delivering fault current long enough to clear a fuse. A fuse is a reasonable choice.
If it's a switching supply, or the load is switched with solid state devices (as opposed to mechanical switches, relays, or hard-wired), then the current is probably limited by one of those elements, and a fuse will not clear (or not reliably).
Most SMPS are designed to fail gracefully into momentary or continuous shorts. (If you have one that is not, please dispose of it and get one that's worth your safety!) No fuse is necessary here.
If the load is solid state switched, you must use a current limited switch, such as the aforementioned protected switch types, or with current limiting circuitry added. A fuse WILL NOT protect a transistor, unless the transistor is obnoxiously oversized for the application (a 1A fuse will easily carry >50A of fault current, so you'd need a 50A transistor to reliably switch and protect that 1A branch).
Power distribution rules still apply. Example: if you have a 20A PSU, whether it's SMPS or not, it's capable of plenty of current. If you then wire up a couple dozen of these strings, each branch only needs to handle the ~1A it consumes, and therefore can use smaller wire. Where that wire connects to the main supply, it must be fused accordingly.
Tim