It depends on what you are looking for in a probabilistic programming system. If you want something that is more of a library than a language you gain the advantages of being integrated into a mainstream language. This would point to systems like Figaro, PyMC, Edward and Anglican. If you don't mind a standalone language you can choose systems like WebPPL, Hakaru, Stan, or Venture. There are tradeoffs in expressivity as well. There are probabilistic models you can express in WebPPL or Anglican that you can't in Stan. Also different systems support different inference algorithms. So if you want to do something like Latent Dirichlet Allocation, I think JAGS still does better than Stan using a naive implementation. At the current time, to use these systems productively, you should have some idea of what model you want to write and given your data what inference methods you expect to work with that model.

I think Figaro, Stan and PyMC are the most "production-ready" in the sense they have been used for projects outside the realms of their creators. Still I would argue on some level all of them are research projects that aim to explore how to make probabilistic modeling more accessible to people. Ideas in one language often will appear in another down the line. So I encourage to explore a few of them and reach out to the people working on them.