What are the "numerator" and "denominator" of binomial coefficients called? Do the numbers $n$ and $k$ in the binomial coefficient $\binom nk$ have a name? 
For the fraction $\frac nk$ we would use numerator and denominator. But I have not seen some terminology for binomial coefficients used anywhere. 
Are some names for these numbers occasionally used? 
I would expect that situations, when you are talking with somebody about some result including binomial coefficients and you need to refer to one of these two numbers, arise quite commonly. For example, when describing Vandermonde's identity 
$$\binom{m+n}r=\sum\limits_{k=0}^r \binom mk \binom n{r-k}$$ 
you could say something like: "Notice that in each summand the sum of numerators of binomial coefficients is the same as on the L.H.S. The same is true for denominators. In the sum, one of the denominator is increasing, while the other one is decreasing, so that their sum remains constant."
 A: 
In Concrete Mathematics, section 5.1 Binomial Coefficients, Basic Identities D.E. Knuth introduces for $n$ the term upper index and for $k$ the term lower index  in the binomial coefficient $$\binom{n}{k}$$
Since he is one of the great who also cares about notation and terminology, we can use these terms without hesitation.

Hint: With respect to carefully considered notational conventions you may find his papers Two notes on Notation and Bracket notation for the "coefficient of" operator instructive.
A: The terms "upper index" and "lower index," while serviceable, are dependent on the particular notational convention $\binom nk$.  For example, in the admittedly unfortunate notation ${}_nC_k$ which appears in some textbooks, "upper index" and "lower index" don't make sense. The primary objection to these terms is that they refer to symbolic (or syntactic) properties, rather than semantic properties.
Instead of "numerator" and "denominator", we can take "dividend" and "divisor" as an excellent precedent.  For binomial coefficients, the analogous terms are "selectend" and "selector" (which are obviously better than "choosend" and "choosor".)  To take it for a spin, let's describe Vandermonde's identity: 
"Notice that in each summand, the sum of the selectends is the same as on the L.H.S. The same is true for the selectors. In the sum, one of the selectors is increasing, while the other one is decreasing, so that their sum remains constant."
