Without many details given about the client, I’d say a basic closed chain movement with very natural mechanics, large range of motion and low injury risk to develop strength, balance and mobility without leaning too heavily on any single one of those criteria. The best movement in that category, in my opinion, that’s easy to teach and has high accessibility, is the goblet squat, done with progressively larger range of motion. Maybe I’m oversimplifying the issue, but sometimes a simple answer works best, at least as a starting place. If this isn’t helpful, share some more details about the client and any imbalances and/or specific weaknesses and/or areas of excessively poor flexibility. Here’s a primer on the goblet squat: http://formfromfunction.com/blog/the-squat-starting-at-the-begining-with…
It’s best to send them to a private physical therapist. About $100 for a visit. They will work with you to come up with some exercises to do.
What can you do as a trainer? Make sure that the muscles and joints below the knee and above are functioning properly.
Foam roll It band, quads, inner thighs, hamstrings, calfs etc. Make sure you get all the knots out. Stretch each muscle after foam rolling it.
Strengthen weak muscles!
Check their feet. Do they have flat feet or high arches. They may need an insert with arch supports in their shoes. This is a big one that is overlooked quite often.
Again it is best to take them to a physical therapist and meet them there at the appointment then you will learn a whole lot of info that will help you with you current client and many future clients as well. You will be seen as professional if you do this and you will not be seen as ignorant if you tell your client that you are a personal trainer and not a therapist. But you can work on your client under a therapists supervision.
I would have them step up and down on a step and incorporate a variety of arm exercises.
I would have them hold on to the rail when they do go upstairs.
I would have them do a variety of quad/hamstring exercises/functional along with isometrics.
As for seeing a PT as stated above, I would first make sure theres nothing else going on with the client, get clearance from their doc.