[Schoen and Yau][1] define a class $\mathcal{C}_n$ of closed smooth $n$-manifolds inductively. $\mathcal{C}_3$ consists of 3-manifolds not containing a surface subgroup. By the geometrization theorem and surface subgroup theorem, this is equivalent to the class of 3-manifolds of positive scalar curvature (equivalently, those which have virtually free fundamental group). Then $\mathcal{C}_4$ is the class of manifolds such that for every covering space, any codimension-1 homology class is realized by a manifold in $\mathcal{C}_3$ or the $\hat{A}$-genus vanishes (which is implied by positive scalar curvature). They prove that a 4-manifold with positive scalar curvature is of class $\mathcal{C}_4$. 

In the case at hand of $Y\times S^1$, $[Y\times \ast]\in H_3(Y\times S^1)$ is a codimension-one homology class, hence must be represented by a 3-manifold in $\mathcal{C}_3$ if $Y\times S^1$ has positive scalar curvature. But if $M\subset Y\times S^1, [M]=[Y]\in H_3(Y\times S^1)$, then there is a non-zero degree map $M\to Y$, and hence if $M\in \mathcal{C}_3$, then $Y\in \mathcal{C}_3$. Hence $Y\times S^1$ admits a metric of positive scalar curvature iff $Y$ does. 


  [1]: http://www.ams.org/mathscinet-getitem?mr=535700