Density of semantics in syntax
In terms of related work, Marek Zaionc and collaborators have been studying similar kinds of questions for some time.  The following paper includes many results:

René David, Katarzyna Grygiel, Jakub Kozic, Christophe Raffalli, Guillaume Theyssier, Marek Zaionc.  Asymptotically almost all λ-terms are strongly normalizing.  Logical Methods in Computer Science, Volume 9, Issue 1 (February 15, 2013). arXiv

The result stated in the title is not exactly the same as the one you are asking about, but related: they prove that the asymptotic density of strongly normalizing terms among all terms of $\lambda$-calculus is 1.  Surprisingly, they also prove that the asymptotic density of strongly normalizing SKI combinator terms among all SKI combinator terms is 0.
As suggested by Gabriel Scherer's response, the specific formulation of density you are asking about is related to the density of normal terms among all terms, or in other words the probability that a uniformly random term of a given size is already normalized.  For subsystems of $\lambda$-calculus the probability that a term is normal typically decreases exponentially with size, and I believe it should be a small step from there to proving that the asymptotic density you are interested in is 0.  Certainly this is true for linear $\lambda$-terms: since normalization cannot increase the size of a term, the number of equivalence classes of terms of size $n$ is at most $n$ times the number of normal terms of size $n$, hence the ratio of equivalence classes to terms is asymptotically 0.