Let $X_1, X_2, \ldots $ be a sequence of independent random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and let $n \in \mathbb{N}$. Let \begin{align} \mathcal{G} = \sigma(X_1, X_2, \ldots, X_n), \qquad \mathcal{T} = \sigma(X_{n+1}, X_{n+2}, \ldots), \\ \mathcal{I} =\{ \{X_1 \in B_1, \ldots, X_n \in B_n\}: B_1 \in \mathcal{B}, \ldots, B_n \in \mathcal{B} \}. \end{align} The aim is to prove that $\mathcal{I}$ is a $\pi$-system that generates $\mathcal{G}$.
Therefore, we take $C, D \in \mathcal{I}$ to show that $C \cap D \in \mathcal{I}$. So, \begin{align} C &= \{ \{X_{i_C} \in B_i \} : 1 \leq i \leq n,\ \ B_i \in \mathcal{B}\} \\ D &= \{ \{X_{i_D} \in B_i \} : 1 \leq i \leq n\,\ \ B_i \in \mathcal{B}\}. \end{align} Now, we want to show that $X_{i_C} \cap X_{i_D} \in B_i \in \mathcal{B}$ for all $1 \leq i \leq n$.
The only thing I can derive from the above is that \begin{align} X_{i_C} \cap X_{i_D} \begin{cases} = \emptyset\qquad \text{when $\ldots$}\\ \neq \emptyset\qquad \text{when $\ldots$}. \end{cases} \end{align} But I do not see when this is the case and how it will be an element of $B_i$.
No, you are doing it wrong conceptually.
To show that $\mathcal I$ is a $\pi$ system, take $C,D\in\mathcal I$. Then your $X_i$ are fixed, they cannot change. What changes is your sets $B_i$.
$C$ should be $\{X_i\in B_i| B_i\in\mathcal B_i\forall i\}$ and $D$ should be $\{X_i\in B_i'| B_i'\in \mathcal B_i\forall i\}$. Do you see the difference? Your random variables cannot change, their codomain sets i.e. $B_i$ should be possibly different in $C$ and $D$.
Thus, $C\cap D=\{X_i\in B_i\cap B_i'|B_i,B_i'\in\mathcal B_i\}$ which shows (since $B_i\cap B_i'\in\mathcal B_i$) that $C\cap D\in\mathcal I$ i.e. $\mathcal I$ is a $\pi$ system.
Can you show it generates $\mathcal G$? You may try to show that $\mathcal G$ is the smallest $\lambda$ system containing $\mathcal I$.