Let $X_1,X_2,\ldots, X_n$ be finitely many topological spaces. For $i=1,2,\ldots, n$, if $\mathscr B_i$ is the base of $X_i$, then it is well know that the family $$\mathscr B=\{B_1\times\cdots \times B_n: B_i\in \mathscr B_i, 1\le i\le n\}$$ is the base of the product space $X_1\times\cdots\times X_n$.
My question is: Does this hold for infinite product topological space? I feel it is not ture. Are there Some counterexamples available?
My question is: Does this hold for infinite product topological space?
Depends on how you define infinite product of spaces. For a given collection $\{X_i\}_{i\in I}$ of topological spaces one way to define a topology on the infinite Cartesian product is to take the following as basis:
$$\mathcal{B}=\bigg\{\prod_{i\in I}B_i\ |\ B_i\in \mathcal{B}_i\bigg\}$$
which is known as box topology. And this is what you refer to. However typically we would take
$$\mathcal{B}=\bigg\{\prod_{i\in I}B_i\ |\ B_i\in \mathcal{B}_i\text{ and }B_i=X_i\text{ for all but finitely many }i\bigg\}$$
which is the product topology.
Of course these two coincide when $I$ is finite, or more generally when only finitely many $X_i$ are not singletons. And they are different otherwise. They also have different properties. For example $\prod_{n=1}^\infty[0,1]$ is compact with product topology (Tychonoff's theorem) but it is not with the box topology.
