What is your description of a "non-slip" shoe? I suggest to mine guys that they wear a shoe that has an arch in the middle and to stand on ladders with the middle of the foot on the rung. I would think "requiring" them to wear a certain attire that is essential to your companies work would be essentially but not legally your responsibility at least to some degree. For instance, I would think if I were to require my guys to wear certain shoes like Korkers that I would at least offer a sum they use towards the purchase or some type of reimbursement over time.
As far as using stoppers all the time. For me, I can't see it. While speed isn't everything it is vital in what you would expect each employee to achieve in time. Using a stopper on every ladder placement would slow down a guy working his way around a house. I have levelers on all my ladders and recently purchased some sectionals and they work great to put you places where other ladders cannot. I use my ladder stopped indoors for all placements with my sectional and outdoors on porches or decks. If the surface below the ladder is dirt or grass I am confident I can secure a ladder placement that I can feel comfortable with.
I do agree though wholeheartedly that ladders are nothing to play with and if using the stopper on every placement makes you feel better about things then go for it!