I can give an example of ring $A$ which is reduced and every non-unit is a zero-divisor. It may be helpful to find the answer over $A$. Let $R=k[x_1,x_2,...]$ be polynomial ring (in infinitely many variables) over a field; let $m =(x_1,x_2,...)$ be a maximal ideal. Fix an integer $n>0$ and define $I$ to be the ideal generated by products $x_l x_j$ with $l$, $j$ distinct and $l> n$. Define $A= R_m /I$. Then $ A$ is reduced, total quotient field of $A$ is $A$, and dimension of $A$ is $n$.
The dimension of a reduced commutative Noetherian ring having no non-zerodivisor is $0$. Hence Tom's comment will not work in general, as commented by others.