I'll mention one more thing which helps: Concrete offers of help. Depending on the circumstances, those can be practical (e.g. cooking and bringing food, covering part of funeral expenses/logistics) or supportive (e.g. making yourself available to talk through things or just be there for someone). A vague offer of help, many people will never make use of. "If there's anything I can do..." often doesn't quite signal "…
"Let me know if there's anything I can do" isn't all that helpful without being specific.