Answer
$x^2=-4y$, $(-2,-1),(2,-1)$,
See graph.
Work Step by Step
1. Given the focus at $(0,-1)$, directrix $y=1$, we have $2p=2$ and $p=1$. We can write the parabola equation in a standard form $4py=-x^2$. With $p=1$, we get $x^2=-4y$
2. Let $y=-1$, we have $x^2=4$ and $x=\pm2$, thus the two points that define the latus rectum are $(-2,-1),(2,-1)$
3. See graph.