How to properly subscript in a superscripted fraction?

\displaystyle won't help you here, since your fraction is in the exponent.

a recommended approach when $e$ is involved is to use the operator name "exp":

$\exp\left(\frac{E_{A}}{k_{B} T}\right)$

\exp is already defined in plain tex and latex.

Two-story fractions in exponents are not recommended; it's better to use the slash form

\[ e^{E_{A}/(k_{B}T)} \]

or maybe

\[ \exp(E_{A}/(k_{B}T) \]

are clearer.

As others have already pointed out, if the code snippet

$e^\frac{E_{A}}{k_{B} T}\frac{E_{A}}{k_{B} T}$

is processed in TeX's inline math mode (as above), you can't get the subscripts A and B to be typeset in an even smaller font size becausethey're already at their smallest possible size. If for some reason you can't follow the advice provided by some of the other answers, I suggest you try the following. A reason the typeset formula is a bit hard to parse/read is that the horizontal space between the E and subscript A, and between the k and the subscript B is quite large. (This happens because both subscripts happen to be uppercase letters with bottom serifs that stick quite out to the left.) This can be remedied nicely by putting a negative thinspace, \!, before each of the subscripts' arguments, as follows:

$e^{\frac{E_{\!A}}{k_{\!B} T}}t\frac{E_{\!A}}{k_{\!B} T}$

IMHO this looks much easier to parse (when it's typeset, of course!).