What fraction of the vapour in electronic cigarettes is water?
The short answer to this is that there is no single answer to this problem as the vapor composition that is produced by any given atomizer is dependant upon a large number of components. To name a few: 


The initial composition of the liquid, which will vary flavor to flavor and manufacturer to manufacturer. You can find out a bit more about this with the following articles 


Characterisation of mainstream and passive vapours emitted by selected electronic cigarettes DOI 10.1016/j.ijheh.2014.10.001
Electronic cigarettes: overview of chemical composition and exposure estimation PMCID: PMC4304610

     A note to consider with the above is that many toxic organic solvents such as Formaldehyde, Acetaldehyde, Acetone, Acrolein, Propanal, Ethylene glycol, and    Diethylene glycol are found in eLiquids, in various concentrations.

The temperature that the eLiquid is heated to internally (this affects the vapor pressures of the various components in the liquid)
The concentrations of all of the, solvents, flavorings and such which are adsorbed by the person who is using the atomizer


Given all of these factors, and the large number that I am sure that I am missing, there is simply too many confounding variables to be able to give a straight answer to your question without directly measuring the concentration - not that I would know how you would make such a measurement.