Solution: We need to show that $\overline{A \times B} = X \times Y $. That is, given any point $(x,y) \in X \times Y$ and any open set $O \ni (x,y)$ must intersects the set $A \times B$. Instead of any open set $O$, we will show that intersection with basic open sets, containing the point $(x,y)$, is nonempty. Take a basic open set $U \times V$ containing the point $(x,y)$. This means $x \in U$ and $y \in V$. As $A$ is dense in $X$ and $B$ is dense in $Y$, we must have $U \cap (A\setminus \{ x \} ) \neq \emptyset $ and $V \cap (B \setminus \{ y \} ) \neq \emptyset $. This implies, \[ (U \times V) \cap \big((A \times B) \setminus (x,y)\big) \neq \emptyset . \]