I think Robins's answer to my question on MO also answers this one.
A descriptive complexity characterization of a complexity class $C$ gives a language whose queries (i.e. formulas) are exactly the functions computable in $C$. The syntax of the language is usually very simple, i.e. given a string $q$ it is easy to check if $q$ is a well-formed query of the language, at least it is expected to be decidable (but usually syntax checking cen be done in a small complexity class). This would entail effective enumerablity of the problems in the class $C$ and would give a syntactic characterization for $C$. (If the complexity of syntax checking is low it might also imply the existence of a complete problem for the class.)
In the comments above, Robin linked to Kord Eickmeyer and Martin Grohe's paper "Randomization and Derandomization in Descriptive Complexity Theory" which gives a "descriptive complexity" characterization of $BPP$. The authors themselves note in the introduction that this is different from what is usually meant by a descriptive complexity characterization:
We prove that $BPIFP+C$, the probabilistic version of fixed-point logic with counting, captures the complexity class $BPP$, even on unordered structures. For ordered structures, this result is a direct consequence of the Immerman-Vardi Theorem [7, 8], and for arbitrary structures it follows from the observation that we can define a random order with high probability in BPIFP+C. Still, the result is surprising at first sight because of its similarity with the open question of whether there is a logic capturing $P$, and because it is believed that $P = BPP$. The caveat is that the logic $BPIFP+C$ does not have an effective syntax and thus is not a “logic” according to Gurevich’s  definition
underlying the question for a logic that captures $P$. Nevertheless, we believe that $BPIFP+C$ gives a completely adequate description of the complexity class $BPP$, because the definition of $BPP$ is inherently ineffective as well (as opposed to the definition of $P$ in terms of the decidable set of polynomially clocked Turing machines).
I am not an expert in finite model theory/descriptive complexity (and personally would like to hear more from experts), but my feeling is that there is a little bit cheating here in saying that this is a descriptive complexity characterization. The reason for my feeling is that if we are allowed to have non-effective syntax, we can use arbitrary semantical restrictions to restrict the class of well-formed queries and can give a "descriptive complexity" characterization for any complexity class. For example, consider $SO(TC)$ (which captures $PSpace$), and then take exactly those queries which are computable in $BQP$; or consider the language that has one function symbol for each machine in $BQP$. Both of these capture $BQP$ but don't have an effective syntax.