This question already has an answer here:
Suppose that $R = S × T$ is a direct product of rings with $S$ and $T$ each having at least two elements. Prove that $R$ has zero divisors.
Hmmm...what about the elements $\,(1,0)\,\,,\,\,(0,1)\,$?
Of course if one, or both, of the rings have no unit you can choose any non-zero elements instead of $\,1\,$
What does it mean for $R$ to have zero divisors? That means it has nonzero elements that multiply to zero. How do you write elements of $R$? as ordered pairs of elements from $S$ and $T$.
So you are asking if there are any non-zero solutions to
$$ (s_1, t_1) (s_2, t_2) = (0, 0) $$
such that $(s_1, t_1) \neq (0,0)$ and $(s_2, t_2) \neq (0,0)$.
That is, you want to find values $s_1, s_2 \in S$ and $t_1, t_2 \in T$ such that all of the following hold: