Fiber optics depend on the surface reflecting light by total internal reflection. Therefore the properties of the outer surface are crucial, and reflection depends on what your understanding of a "stream" is.
If the water is in direct contact with anything solid (or liquid) (most likely anything but air), it will not work. Total internal reflection depends on the outer material being transparent (for the wavelength considered) with the refractive index being smaller than that for water (1.333).
If your surface is not smooth (having air bubbles or the like), you will most likely get a significant intensity drop of transmitted light. Detection of a broken stream in such cases will depend on the power of your light source, the sensitivity of your detector and the length of the stream.
As long as the surface is smooth, it should work. However, my everyday life experience with water tells me that freestream liquid jets tend to become unstable and separate into drops. I presume there are a lot of publications out there on this phenomenon. This is something that would need further investigation.
Also, a curly surface might cause troubles. I don't know if there are any investigations on this, but you should have a look into this.
Regarding your comment on turbulence: Turbulence alone does not influence transmisison of light in water. However, it influences the outer water surface of a free stream. Nice pictures of this process are here: http://dx.doi.org/10.1016/j.ijmultiphaseflow.2013.11.013 (open access article). They show that turbulent free streams are unstable and disperse into small drops.