I used to work in liability insurance claims, and certainly the question of whether or not the man should have been delivering letters to your house is irrelevant - he was there, and the incident arose, that's the sum total of it.
If it is on your property and you have an obligation to keep the step safe, then you could be deemed to be negligent, but - and this is a big but - you need to be aware of the condition of the step in the first place, and know whether or not it's your responsibility. Don't do as others suggest and run round and repair it - repairing or remedying something straight after an accident is about as clear an admission of liability as you can make.
What I suggest you do is go now and take photos of the offending step and go to the council offices and speak to the Highways department, show them the photo and ask for clarification as to whether it's part of the public footpath or part of your house. Don't tell your insurance company yet - after all, nothing may come of it - and the man has three years to claim (it's three years after the date of the incident, or for personal injury claims, three years after he became aware of any long-lasting effects following his fall) but PLEASE write it all down now, while it's fresh in your mind. That way, if a claim comes in in six months' time, you'll have all the information to hand, and can send it straight to your household insurers (you don't need to reply other than telling him in writing who you are sending it to - no apologies, just acknowledgement of the letter). Your household insurance company will appoint solicitors to deal with the claim on your behalf, it won't cost you for their services. If you are asked why you didn't report it at the time, you can say it's because the man said he was OK.
..and in answer to questions about who leaves letters sticking out of the door, it happens round here often, particularly if you have a stiff letterbox.