If you are walking with an unsteady patient, how can you best prevent a fall?

No! Do not hold the person by their arm! You dislocate more shoulders that way.

The best thing to do is to get a qualified and experienced Physical Therapist first to show you how to support the person, then to watch you doing it and correct your technique.

You need to get up close and personal. You stand hip to hip with the person, and you place your arm behind their back, around their lower waist, and hold on to their opposite hip so that you are holding the person actually up against you. This prevents any stumble developing enough momentum to become a fall before you can get a firm hold. The person then holds on to your arm, and off you both go, as with a three-legged race.

If the person is using a walker, you may prefer to use a gait belt. This, again, fits around the person's lower waist/upper hips rather than waist as such, and has handles which you can hold firmly as you stand behind the person.

It is of no use trying to catch a person who has already started to fall. You're most unlikely to have the physical strength or speed to be able to do it, and you may injure the person in the attempt. That's why your hold needs to be firm enough to prevent their weight overtipping their centre of gravity in the first place.

There is also a lot to be said for a wheelchair, of course :)

Unfortunately no matter what you do or how careful you are, falls happen.  That is the way gravity works.  Also, sometimes the bones of an elderly person are so brittle that the bone breaks while the person is standing or walking thus causing the person to fall.

In regards to your husband's fall, since his fall was due to the walker hitting a step, try to watch where your husband is placing his walker and attempt to anticipate any problems.  DO NOT GRAB for the walker as it falls forward because the weight of your husband and the leaning walker can cause you to lose your balance and to fall also.  (I had a CNA who grabbed the walker from behind as the resident was falling forward and she fell forward also landing on top of the resident and their walker.) 

 Always use a gait belt with someone who is unsteady on their feet (even when the person is using a walker--  That way you have something to hold to so that you can help steady the person if they are having trouble with the walker.)  If a gait belt is unavailable, then you can hold on the waistband of their pants or slacks while they walk.
Contact your local Physical Therapist and ask to be shown how to use a gait belt on your Loved One.

Google "How to use a gait belt" and you will find several YouTube videos on how to use a gait belt.

Copy and Paste the URLs listed below to your web browser for some videos and PDFs on "How to use a gait belt". 

https://www.youtube.com/watch?v=SaGHn0JOpkQ
"Using a Gait Belt Safely and Effectively On Any Size Person"   shows how to correctly put a gait belt on a person.

https://www.youtube.com/watch?annotation_id=annotation_947055&feature=iv&src_vid=OBVqxwggZ3U&v=_irfJdCL2Dw
"Transfer from Bed to Wheelchair CNA Skill NEW" shows how to transfer a person from a bed to chair.  (I brace one or both of my knees against the patient's knees to prevent their knees from buckling during the transfer.)

http://www.med.umich.edu/1libr/FallsPreventionCommittee/UsingAGaitBelt.pdf
"Using a Gait Belt from University of Michigan" pamphlet

https://www.caregiver-aid.com/ambulation-gait-belt/

http://www.elder-home-care-help.com/gait-belt.html

Hope that this information is helpful.

You need a Gait Belt.
Once the gate belt is on you walk to the side and a bit behind the person with your hand on the gait belt at all times.
If the person begins to fall you can get behind them and position yourself so you can guide them down safely Do NOT try to prevent the fall or you both may go down. Once they are safely down then you can try getting them up but if you can not do it safely call emergency service and request a "lift assist" there will be no transport to the hospital so in most areas there will be no charge for the lift assist. You may have to sign a paper stating that there was no request for transport to the hospital.
The gait belt should be positioned between the waist and chest
It should not be loose enough that it will slide around and you need to get your 4 fingers under it and be comfortable. It should not be so tight that it is uncomfortable for you to keep your fingers in place and the person you are guiding should be comfortable.
There are different kinds I would get one that is wide and sturdy. I had two, one wider and heavier duty and a thinner more light weight one. The heavier, wider one was more comfortable and easier to get one and buckle.
When buying one the person selling should be able to show you the correct way to put it on. I wrote my Husbands name on the belt on the inside so I could quickly tell at a glance what was the inside and if it was upside down or not. Made getting it on faster and easier.

Use a gait belt or hold onto the waist of their pants. That's how we do it at the nursing home.

Echoing Countrymouse's advice. When Mom became really unsteady, I tried a gait belt on her. Magic device. Mom could walk without anyone hanging on to her, which she hated. She hated the walker too. She seemed to feel more free, but if she staggered--in any direction--holding her by the belt steadied her until she could get her balance. (I really appreciate CM's recommendation for the PT, as there may be something particular with the client that your friend should do or avoid in helping her client.)

There is a belt available for this purpose. You put this on the patient and hold on to the belt. Walk behind holding to the belt tight. That is how they trained me after my open heart surgery. I use this to walk with my mother in law. Check with a PT or durable medical supply store

Lot's of great advice and methods here. I just wanted to be sure to add, falls can and are likely to happen even when you are doing everything right and being diligent. You will minimize them and can very well minimize them, really what you are there doing is making a fall less damaging, easing them to the ground rather than "preventing" or catching them from falling (you don't want to hurt yourself in the process or your both screwed) and you should never feel guilty or responsible for falls or injuries when you are taking all the precautions. Even when something happens because you didn't happen to be in your regular spot or helping your LO from the walker to the bed or down the hall in the house because they decided to do it on their own, don't let yourself feel responsible or beat yourself up. No "if only's" because falls and accidents happen, sometimes for a bigger reason and sometimes just because they do, it's the nature of life. Good for you for educating yourself in every way you can you re obviously very diligent, aware, concerned and I feel very confident in saying it wont be your fault! :)

If the person can't walk safely, then you shouldn't be the prop. Very few people are stable themselves if the weight of another adult falls on them or pulls away from them. For guidance for the blind or vision impaired who are stable on their own feet, the method is normally for them to put their own arm through yours as you walk side by side. Was your husband's fall a sheer accident? or caused by low vision? or is he past a walker and needs a wheelchair? Think first of the safety or the helper. If the helper falls and fractures something, they can't be the helper any more, so they are important themselves.

If your elder will allow it, use a gait Belt. It goes around the elder at waist high. You hold the belt in back. Helps to hold on to the elder in case of fall and getting them up after a fall.

Ask your family doctor for "home health help". The therapist is is an excellent resourse and will give you practical information and how to use a gait belt if that's an option.

Or contact the local hospital to see if the have people to advise you.

You said it yourself, the key is close supervision. Many people with dementia have trouble using walking devices because they can't remember how. If someone is using a walker, stand behind them with your hands at their waist. If you don't have a gait belt, use any belt with a secure buckle. If you grab them by the arms or armpit, you upset their natural mechanism.