Reachability is unlikely to work. For a DFA/NFA, there are only finitely many states to consider but, for a Turing machine, there are infinitely many possible configurations of the tape and state.
Instead, you need to simulate the machine you're interested in on possible inputs and see if it accepts any of them. In principle, you'd want to try each possible input in order (e.g., $\epsilon$, $0$, $1$, $00$, $01$, $10$, $11$, $000$, ...) but there's a problem with this. If the machine doesn't halt on some input, you'll never begin to consider the next input. You get around this with a technique known as "dovetailing". You simulate your machine running for one step on the first input. If that's enough to work out the answer, you're done. Otherwise, you simulate the machine running for two steps on each of the first two inputs. If that's not enough, three steps on the first three inputs, and so on.
I'll leave it to you to check that this technique really works and to use it to answer your question.