H
|z- 3i| is the distance from z to 3i and |x- 2| is the distance from z to 2 so S consists of points that are equally distant from 3i and 2. Geometrically, the set of all points equally distant from point P and Q is the perpendicular bisector of the segment PQ.
Algebraically, taking z= x+ iy, |z- 3i|= |z- 2| is the same as $\displaystyle \sqrt{(x^2+ (y- 3))^2}= \sqrt{((x- 2)^2+ y^2}$ which is the same as $\displaystyle x^2+ (y- 3)^2= (x- 2)^2+ y^2$.
As for whether it is open or close, what topology are you using?