Let $E/F$ be a quadratic extension of number fields and $v$ is a place of $F$.
Let $\chi_1,\chi_2$ be the unramified characters of $F_v^{\times}$.
If $B(\chi_1,\chi_2)$ is the unramified principal series representation of $GL_2(F_v)$, what is the $BC(\pi)$, the base change of $\pi$ to $GL_2(E_v)$?
I suppose that $BC(\pi)=B(\chi_1 \circ \text{Norm}_{E_v/F_v},\chi_2 \circ \text{Norm}_{E_v/F_v})$. Is this right?
The answer to your final question should be yes.
Let me assume that $v$ is inert in $E$, so that $E_v / F_v$ is a honest quadratic extension. Assuming that $B(\chi_1, \chi_2)$ is irreducible, it corresponds via local Langlands (a theorem for $\mathrm{GL}_2(K)$ and any $p$-adic field $K$) to the 2-dimensional representation of the Weil group of $F_v$ given by the sum of the characters $\chi_1$ and $\chi_2$.
The operation of base change on the $p$-adic side corresponds to restriction of Galois representations, thus the base change of $B(\chi_1, \chi_2)$ corresponds via local Langlands to the restriction of $\chi_1 \oplus \chi_2$ to the Weil group of $E_v$.
When we restrict $\chi_1 \oplus \chi_2$ to the Weil group of $E_v$ and then make this homomorphism factor through the abelianization $E_v^*$, local class field theory tells us that this corresponds exactly to pre-composition of $\chi_1$ and $\chi_2$ with the Norm map. Now go back to the $p$-adic side and you have your statement.
