I always take the time to consider how I can help others.

"Let me know if you need anything" puts the work back on the person least able to do it. Someone in the middle of a bad month now has to identify a task, judge whether it is too big to ask for, and compose the request.

Considering how you can help means doing that thinking yourself. Their bins go out Tuesday. Their car needs its MOT. Nobody has cooked in that house for a week.

Then you offer the specific thing, which is far easier to accept and much harder to refuse out of politeness.