My understanding of the mind/body problem is that the relationship between the mind and body can be perceived in one of two ways. The first is called the "monistic" theory that states that the mind and body are not separate.
This view was first advocated in Western philosophy by Parmenides in the 5th century BC.
The second is the dualistic theory, which can be traced back to Plato and Aristotle. According to this way of thinking, the mind and body were not one, but two separate parts:
...mind is thought to be of a substance other than a physical substance
I would assume that Shakespeare disagreed with the monistic theory, and agreed with the dualistic precept.
In Macbeth, there is reason to believe that Shakespeare sees a separation between mind and body.
Macbeth finds that Macduff has fled to England. He reminds himself that from now, on, he will make sure his thoughts and actions are completed at the same time, inferring that they have been separate before now:
From this moment
The very firstlings of my heart shall be
The firstlings of my hand. (IV.i.46-48)
Also in Macbeth, when Lady Macbeth begins to walk in her sleep, her mind is addled and her dreams are haunted, yet she gets up and walks like she would when awake, wringing and washing her hands. It would seem that the mind is not in tact, but that the body still functions.
In this play, the mind is often at odds with the body. Again, in the dagger scene, Macbeth's mind perceives the dagger, but his hand cannot grasp it.
In Hamlet, this theme is also seen. Hamlet considers suicide but is concerned that instead of nothingness after death, there might be dreams to torment one. Dreams are a part of the mind; if the body is dead, the mind must be separate to survive and have dreams.
In my opinion, Shakespeare would seem more a dualistic theorist rather than a monistic theorist.