I've trained some people with vertigo, as well as many with balance issues.
One approach I take is trying to get them stronger while not worrying about balance, with the idea that down the line the extra strength will make it easier to then work on the balance. Basically my biggest concern with anyone mentioning balance right off the bat is making sure they don't fall. If the person is mentioning it right away, it's usually pretty bad.
For instance, rather than play with squatting with a barbell, I just have them Leg Press. This way we don't have to worry about balance, or losing the bar at all.
For lunging, I'll have them lunge in place, while holding on to something. (Over time you can play with not holding on, but keeping a hand nearby just in case.)
Sled work is also great.
Next, I try to minimize how much up and down motion we do. I don't have them get up and down off the ground much, or if we do anything where the person leans over, I tell them to keep their eyes up so at least their head isn't looking down and up.
I've found many, when going from horizontal to vertical fairly quickly, get very thrown off.