What's actually going on
Rust-colored water at your downstairs fixtures, and nowhere else, is your plumbing pointing you at the lowest part of the system.
Two things cause this most of the time: corroded galvanized supply pipe, or a water heater full of sediment feeding the lower level.
The reason it's downstairs and not upstairs is simple. Rust and mineral sediment are heavy. They settle into the lowest horizontal pipe runs and sit there, and old steel pipe rots from the inside out starting at the bottom.
Here's the quick test that tells you a lot. Run the cold tap for two to three minutes. If it clears and stays clear, an outside event stirred things up. If it clears then comes back brown within a day, your pipe is actively shedding rust, and that's a material problem you can't clean away.
The most likely culprits, ranked
Start with the water heater versus pipes question, because it splits the whole diagnosis in half.
If only your hot water runs rusty downstairs, the heater is almost certainly the source. Tanks collect rust and mineral sediment over the years, and the lowest branch off the heater catches the worst of it.
If your cold water runs rusty, or both hot and cold do, the supply pipes are the problem. In homes built before the mid-1970s, that usually means galvanized steel that's simply worn out.
Galvanized pipe has a zinc coating that protects the steel. That coating wears away over 40 to 60 years, and then the bare steel rusts. Technicians have cut open 3/4-inch galvanized that had corroded down to a pencil-width opening inside, packed with rust scale.
A localized cause is worth ruling out too. A single corroded shutoff valve or steel supply line under one sink can dump rust into just that fixture. If only one tap runs brown and the rest of the floor is clear, that's your answer, and it's a cheap fix.
And don't forget the neighborhood. If a utility crew flushed a hydrant or fixed a main nearby, rust gets stirred into the city line and shows up at your low fixtures first. Ask a neighbor. If their water's off too, wait it out.