For a flow over a smooth sphere, does the wake region increases with increase in Reynolds number? And if so, then why?
Before the discussion of a sphere, I would like to mention how the flow across a long cylinder (i.e. a circle in 2 dimensions) progresses (and why so) with an increase in Reynolds number (Re).
Consider a flow across the cylinder in the creeping flow regime ($Re\leq 1$). This means that the inertial forces are low compared to the viscous forces. Consider what this implies. The inertial forces cause the fluid particles (and effectively the flow) to continue in its state of uniform motion. If the inertial forces are lower than the viscous forces, the viscous forces will dominate and the fluid particles do not continue in their state of uniform motion; which would be a tangent to the sphere at any point. Instead, they allow the viscous forces to manipulate the direction of motion, so that the fluid particles skirt along the curved surface of the cylinder. Thus in this regime, no wake is noticed and the flow is symmetric about the vertical diameter of the circle.
As the Re increases, the inertial forces start to increase causing the fluid to become more "stubborn" against the viscosity. This is primarily the reason for the flow to separate tangential to the cylinder. The point of separation depends on how dominant the inertial forces are as compared to the viscous forces; i.e. Re. With an increasing Re, the point of separation is drawn more upstream; causing larger wakes (which is quite easy to imagine).
This behavior continues till the point where the flow becomes turbulent. This is generally known as the "Drag-crisis"; where the drag-coefficient of the cylinder suddenly drops. Turbulent flow is more random in the sense that the momentum transfer in the fluid particles is randomly scattered in all directions; which causes the flow to lose some of its directional nature (which we earlier attributed to the inertial forces). Due to this, the viscous forces are assisted, causing the flow to reattach and get separated further downstream, resulting in a smaller wake in the process. The drag-crisis occurs at $Re=10^5 - 10^6$ for a long-cylinder.
After the drag-crisis, as Re is further increased, the separation begins to be drawn upstream again, causing an increase in the wake-size.
A similar reasoning will lead us to understand that the wake behind a sphere is definitely affected by the Re. And that too in a similar fashion. The increase in Re initially causes a growth in the size of the wake, and after transition to turbulence, a sudden reduction. This is again followed by an increase in the wake-size as Re is further increased.
I would strongly suggest you to check out An Album of Fluid Motion by Milton Van Dyke. It has interesting pictures both these flows, and many others, with good descriptions. It gives a good physical feel of the fluid flow in question.