A rusting anchor chain isn't just a cosmetic problem for your boat. Rust can weaken the chains, eventually causing the links to weaken and break. Clean your anchor chain at least once every year and store it in a dry, clean location to protect it from rust. Although rust can be brushed from an anchor chain, replacing rusty chain and re-galvanising the entire chain is usually the best approach.
Inspect the anchor chain for signs of rust. Anchor chains tend to pick up a considerable amount of debris underwater, so it's best to clean as much as possible first.
Brush the chain with a brass brush, since steel brushes can leave deep scratches in the chain that will lead to more rust.
Ask your local marina or boating supply store for a company that specialises in re-galvanising chains. A galvanised chain will be much more resistant to rust.
Use a pressure-washer to remove debris and other foreign objects from the chain after using the anchor. Store the chain in a clean, dry area.