To what extent has the rise in the influence of the members of the executive office affected our system of checks & balances?
I would argue that the rising influence of the members of the Executive Office of the President has to some extent undercut the checks and balances that were written into our system.
The reason for this is that the EOP is not really subject to any kind of Congressional oversight. You can argue that the Framers intended for the executive branch to be run largely by the members of the Cabinet. These people would be approved by the Senate and that would provide some amount of a check on the executive branch.
With the rise of the EOP, you have much more policy being made and carried out by people who are not subject to being approved by Congress. This diminishes Congress's ability to check the power of the executive.