'Steer clear of xx' is a common idiomatic phrase meaning to avoid or keep away from something or someone.
It's easy to understand if you think about its literal meaning. Imagine that you are steering a boat and you know there are some dangerous rocks nearby. If you steer clear of the rocks, you manoeuvre your boat in a wide arc to make sure that you keep a safe distance away from the rocks as you sail past the area where the danger is.
Likewise, wealthy people made sure that they kept away from the supposedly unhealthy air of hospitals.
So, in fact, we have two natural everyday phrases here: 'keep away from' and 'steer clear of', both meaning to avoid. If you want to intensify them, you could add 'well' e.g. 'keep well away from..' and 'steer well clear of x'...
I hope that's helpful.