How to determine the amount of barium in a sample of contaminated water?
Because the client is ultimately interested in removing the barium by precipitation, gravimetric analysis is likely a suitable solution. This can be done in a few hours in a lab with basic equipment like a filter flask, a drying oven, an analytical balance and the appropriate reagents.   

You can take advantage of the fact that barium sulfate is very insoluble in water by adding some saturated solution of sodium sulfate to a known volume of a sample of the water. Then just filter out the barium sulfate onto some pre-weighed filter paper, dry, weigh, and calculate the mass of barium (from the measured mass and the molecular weight of barium sulfate) per volume of water. You should repeat the analysis to be sure you added enough sodium sulfate solution to precipitate all of the barium.  

If there is little or no precipitate found, you have a couple options. Firstly, you can optimize the gravimetric analysis by using large volumes of sample with a small filter and a good analytical balance. Also be sure to pre-filter the sample prior to adding the sodium sulfate in case there are suspended particulates, which would get measured just as if they were barium.  

If a gravimetric analysis does not provide the detection limit required by the client, the next best options are atomic absorption spectroscopy (AAS) and inductively coupled plasma - atomic emission spectroscopy (ICP-AES). As these analyses can be done very cheaply (~\$10-$30 per sample) you may actually want to do this to begin with. The down side of outsourcing is that the turnaround time could be weeks, or the price could quadruple if you need results fast.