Let me give the answer for $n=2$.

Fix a point $p \in C$ and call $x_1, \, x_2, \, \Delta$ the divisor classes

$\begin{equation*} \begin{split} x_1 & := \{(p, \, x) \, | \, x \in C\} \\ x_2 & := \{(x, \, p) \, | \, x \in C \} \\ \Delta & := \{(x, \, x) \, | \, x \in C \}. \end{split} \end{equation*} $

Let $C^{(2)}$ be the second symmetric product of $C$, namely the quotient of $C^2$ by the action of $\mathbb{Z}/2 \mathbb{Z}$ exchanging the two factors, and let $x$, $\delta$ be the images of $x_1$ (or $x_2$) and $\Delta$ in $C^{(2)}$, respectively.

It is a classical result that for a general curve $C$ of positive genus the Neron-Severi group of $C^{(2)}$ is generated by $x$ and $\delta$, hence a basis for the Neron-Severi group of $C^2$ is given by $$x_1, \, x_2, \, \Delta.$$ The same result holds for the Neron-Severi group of the $n$-fold symmetric product $C^{(n)}$, hence one shows that the Neron Severi group of $C^n$ is generated, for general $C$, by the coordinate divisors and by the classes of diagonals.

References. A. Kouvidakis: Divisors on symmetric product of curves, Trans. Amer. Math. Soc. 337 (1993), 117-128.

Let me give the answer for $n=2$.

Fix a point $p \in C$ and call $x_1, \, x_2, \, \Delta$ the divisor classes

$\begin{equation*} \begin{split} x_1 & := \{(p, \, x) \, | \, x \in C\} \\ x_2 & := \{(x, \, p) \, | \, x \in C \} \\ \Delta & := \{(x, \, x) \, | \, x \in C \}. \end{split} \end{equation*} $

Let $C^{(2)}$ be the second symmetric product of $C$, namely the quotient of $C^2$ by the action of $\mathbb{Z}/2 \mathbb{Z}$ exchanging the two factors, and let $x$, $\delta$ be the images of $x_1$ (or $x_2$) and $\Delta$ in $C^{(2)}$, respectively.

It is a classical result that for a general curve $C$ of positive genus the Neron-Severi group of $C^{(2)}$ is generated by $x$ and $\delta$, hence a basis for the Neron-Severi group of $C^2$ is given by $$x_1, \, x_2, \, \Delta.$$ The same result holds for the Neron-Severi group of the $n$-fold symmetric product $C^{(n)}$, hence one shows that the Neron Severi group of $C^n$ is generated, for general $C$, by the coordinate divisors and by the classes of the diagonals.

References. A. Kouvidakis: Divisors on symmetric product of curves, Trans. Amer. Math. Soc. 337 (1993), 117-128.