Why do wider double wells have a lower $\Delta E$ than thinner ones? 
In this diagram, in which an $n=1 (E_{1})$ and $n=2 (E_{2})$ wave have been superimposed, the probability density of the new, combined wave changes with time. The period of its shifting density is given by $\frac{1}{f}$ where $f = \frac{E_{2}-E_{1}}{h}$ This much is clear to me.
Now a blurb claims: 

A double well with a high or wide barrier will have a smaller $\Delta E = E_{2} - E_{1}$
  than one with a low or narrow barrier. (Less coupling.) 

Why is this? I would reason that $E_{2}$ and $E_{1}$ are inputs in this scenario. We "control" them -- how could changing the barrier's dimensions affect them?
 A: We don't control the allowed energies $E_i$ independently of the potential: the energies must be the eigenvalues of the Hamiltonian.  The "inputs" are the shape and height of the barrier between the two wells.
You can kinda sorta think of the energy difference between the symmetric state (with energy $E_1$ in your diagram) and the antisymmetric state (with energy $E_2$) as the cost for particles trapped in your two wells to have different phases.  If the potential barrier is very high or very wide, as if your two wells were a thousand miles apart, then there's no cost for particles in the two wells to have phases different from each other.  If the barrier is low or thin, then tunnelling means your particles will spend time in both wells and the relative phases are constrained.
Your text should also have given you a derivation and expression for the energies $E_i$ in terms of the shape of the potential; understanding those results, or producing them on your own, should be illuminating.
A: 
A double well with a high or wide barrier will have a smaller $\Delta E=E_2−E_1$ than one with a low or narrow barrier. (Less coupling.)

I think we can understand this intuitively as follows but first it has to be said that rob is right: the energies $E_i$ are NOT inputs but the eigenvalues of the Schrödinger equation. Width, height and potential of the wells are the inputs here.
Now consider a system where $d \approx 0$ (below, left), or at least much, much smaller than the width of the two wells. Such a system would resemble the simple potential well (without central barrier) and would have large separation between the eigenvalues $E_i$, as shown below and calculated on this page.

As we introduce the potential barrier $d$ and widen it (above, right), the difference between eigenvalues $E_1$ and $E_2$ would start to decrease, so wider barriers will result in a smaller difference between $E_1$ and $E_2$.
For a rigorous proof one would have to determine the eigenvalues of the relevant Schrödinger equation(s).
In addition we can say that the 'pairs' $E_1, E_2$ (etc) are likely to lie higher for the double well than $E_1$ for the single well, because narrower wells have higher eigenenergies than wider ones (all other things being equal).
A: Electrons are Fermions so they are forbidden from being in exactly the same state. Two identical quantum wells placed an infinite distance apart will be identical because the wave functions do not overlap. However at the quantum wells are moved closer together the wave functions begin to overlap and the exclusion principle forces the energy levels to split such that they remain unique.
For example, the band structure of semiconductors follows directly from this principle. There we have a huge number of atoms that are brought together which results in conduction and valence bands (formed from individual, originally identical),  atomic orbitals that have shifted in energy.
