“Let me know if you need anything” is kind, but it leaves much of the work with the person who needs help. They must identify a task, estimate what you might be willing to do and decide whether asking would be too much.
Offer something concrete when you can. “I can bring dinner on Wednesday” or “I have an hour to help pack on Saturday” gives the person an option they can understand. Check their preferences and circumstances rather than deciding what they need without asking.
Be accurate about your availability. An offer made from immediate emotion may become difficult to sustain once the practical demands appear. A smaller contribution you can reliably make is often more helpful than a broad promise followed by uncertainty.
Leave room for the person to decline. They may have another arrangement, different needs or a wish for privacy. Help should not require accepting your preferred version of being helpful.
If the situation needs professional support, recognize the limits of your role. Friendship and practical assistance can matter without making you responsible for solving a medical, legal, financial or safety problem beyond your knowledge.
Once an offer is accepted, confirm the details and follow through. If circumstances change, communicate early and help clarify the new arrangement where reasonable. Reliability is part of the kindness.
Be specific about both the task and the limit. “I can bring dinner on Tuesday” is easier to rely on than “Anything you need, anytime.” If you can help for an hour, say so before the work begins. A person dealing with a difficult week may not have the energy to translate a broad offer into an appropriate request, especially if they are worried about asking too much.
Follow through on the version you offered. If circumstances change, communicate promptly so that the person can make another arrangement. You can also ask whether your proposed help would actually be useful instead of assuming. The best intention may not match the need. A smaller, well-matched contribution often feels more supportive than a grand promise that is hard to schedule, unclear in scope or quietly withdrawn when the moment arrives.
You can also state a boundary without withdrawing all care: “I cannot take that on, but I can do this part.” The alternative may or may not meet the need, and the other person gets to say so. A clear offer respects both people's circumstances. It makes generosity something that can be used, rather than an unlimited phrase neither of you knows how to translate into an ordinary Tuesday.
A little more room to be yourself.