What is the descriptive complexity of the basic modal logic? This is kind of multiple questions in one. Let's consider $\phi$ a formula in the basic modal language (that is, in propositional language based on a set of propositional variables $\Phi$, plus one of the basic modal operators, say, $\Diamond$)

1) Define the $\phi-\mathsf{MODEL}$ problem as: Given a finite structure $M = (W, R, V, w)$, where $W$ is a set, $R \subseteq W \times W$, $V : \Phi \rightarrow 2^W$ and $w \in W$, is $\phi$ satisfied in $M$, as per the usual modal logic definition of satisfiability?

2)  Define the $\phi-\mathsf{FRAME}$ problem as: Given a finite structure $F = (W, R)$, where $W$ is a set and $R \subseteq W \times W$ is $\phi$ valid in $F$, as per the usual modal logic definition of validity?

We could go on and define other variants, asking if there is a world in which the formula is satisfied (or valid), if there is a valuation such that the formula is satisfied, and so on.

Now consider, for example, the collection of $\phi-\mathsf{MODEL}$ problems, for all basic modal formulas $\phi$. My question could be formulated as thus: Is this class of problems equal to any well-known complexity class? Of course, the question could be formulated for any of the variants i just proposed, so it's as i said earlier, kind of multiple questions in one.

(It goes without saying, but for other logics, such as first and second-order logics, this is a much-studied subject with the name Descriptive Complexity, with classic results such as Fagin's Theorem relating the existential fragment of second-order logic to the complexity class NP)
Modal logic does not capture any interesting complexity class. As it is invariant under bisimulation, it cannot even test whether the vertex $w$ has degree greater than $2$, which is a constant time property. We are interested in ML for a different reason: ML is the bisimulation invariant fragment of first-order logic (Theorem of van Benthem). In other words, ML is the "first-order logic of process behaviour". Similarly, the bisimulation invariant fragment of monadic second-order logic is the $\mu$-calculus (Theorem of Janin/Walukiewicz).