Do they have to reproduce to maintain a population from when Zeus created it and present day? If so, who are the fathers?
In the film we only see one young Amazonian, Diana and we got to know in the end that she is
the daughter of Zeus
and rest of the Amazonian seems to be older than her and we can assume all were created by Gods. So I think they didn't reproduce and they are all immortal (until killed) as their comics counterpart.
But in comics they did reproduce:
In The New 52 :The Amazons are said to have infrequently invaded ships coming near their island and forced themselves on the sailors, before killing them. After nine months, the birth of the resulting female children was highly celebrated and they were inducted into the ranks of the Amazons while the male children were rejected. In order to save the male children from being drowned to death by the Amazons, Hephaestus traded weapons to the Amazons in exchange for them.
But it isn't mentioned in films. So it will be just assumptions for now.
It's specifically stated in the movie that Diana is the only child ever "born" on Themiscyra.
We also know that the Amazons are essentially immortal although they age but really slowly. We see Diana grow from a young child but we don't actually know how long that takes.
So do they reproduce in the DCEU?
...at least not without divine intervention.
No, the Amazons of Themyscira have no need to maintain the population as they do not age.
As told to Diana by her mother Hippolyta, the Amazons were created by Zeus as an army to defend mankind from the corruptions of Ares. After casting Ares from Olympus, Zeus then created Themyscira for them as a place where time & disease could not touch them. And lastly, we know that Hippolyta and her sister Antiope took part in the original battle against Ares' forces, and have not visibly aged since that time. Therefore, it is safe to assume that the Amazons on the island are the original ones created by Zeus.
This matches the comics version of Themyscira prior to the "New 52" reboot, with most of the Amazons on the island being alive since the days of ancient Greece. They did, however - from time to time - adopt girls into their culture as well.