Recognizer to check if the language of a Turing machine contains a finite subset Let $B = \{ 123 \}$. 

Note that $B$ is finite.

Let $L = \left \{ \left\langle M \right\rangle | M \text{ is a Turing machine such that } B \subseteq L(M)  \right\}$.

Is it sufficient to show that $L$ is recognizable by checking if $M$ accepts on input $123$, and therefore $\left\langle M \right\rangle \in L$?
Yes, it is sufficient. 

$\langle M \rangle \in L \Longleftrightarrow B\subseteq L(M) \Longleftrightarrow M\text{ accepts all strings in }B \Longleftrightarrow M\text{ accepts }123$.