When bad things happen to our friends and family, we often want to help. Redditor usapeaches argues that saying "let me know if you need anything" is not the right thing to say, because you're actually burdening them with the need to reach out to you for help. When you want to offer your assistance, be specific.
When your friend loses a loved one, your sister ends up in hospital or something else bad happens, it's important to be specific about how you can help. In these instances, those in need are in a place where they can't handle their normal day-to-day life on their own, and they aren't going to have an easy time asking you to do them a favour. Offer to watch their kids, pick up groceries, cook them food or walk their dogs. Think about what you can do to help them, rather than task them with assigning you -- and everyone else -- who offered.
While there's nothing inherently wrong with letting someone know you're there to help, if you really want to make yourself useful, then be sure to offer specifics. When you need their help someday, they'll remember.