The short answer to the actual question asked: you might do it electrostatically, but it would require huge amounts of power.
The extended short answer: the water, though a nearly incredible mass of it, is spread over such a gigantic region that it is far less concentrated than in the air of the Atacama Desert, the driest place on Earth.
The doubly extended short answer: the water cloud observed isn't pure water. It will have lots of other not-so-potable stuff, like methane, ammonia, and hydrogen sulfide, mixed in with it.
OK, electrostatic "mining". I've used the Caltech Submillimeter Observatory (CSO) before and I know how they use it. The fact that the observers detected the water at millimeter and submillimeter wavelengths with the CSO tells me that the quasar is irradiating the water, exciting the molecules and causing transitions in their rotational-vibrational modes. Each such transition emits electromagnetic energy at a specific frequency. If you tune the CSO's receivers to 2 or 3 such frequencies specific to water and you see emission or absorption at all of them, then you have a pretty good identification that the stuff doing the emitting or absorbing is water. But quasars emit lots of radiation at energies much higher than mm or sub-mm, and just like UV photons from our sun ionize Earth's upper atmosphere, photons at those higher energies can ionize the water molecules (and everything else, too) in the cloud. They can knock a hydrogen nucleus (a bare proton) from a molecule, leaving H+ and OH-. The H+ often attaches to a neutral water molecule, making H3O+. The net result is that a lot of the water is ionized.
If you place electrodes in the cloud and charge them, H3O+ and H+ will be attracted to the negatively-charged electrode, while electrons and OH- are attracted to the positively-charged electrode. Bringing the H3O+ and OH- together (my guess would be that you do this by using magnetic fields to channel their motion) will yield two molecules of neutral water, which you can collect on a super-cooled surface. Bringing H+ and OH- together give one water molecule. When enough has been collected, you enclose that surface in a suitable container and heat it, driving the water into the container for harvesting.
Recombining those ions releases a fair amount of energy (in the form of photons and kinetic energy of the reaction products) which will tend to heat the apparatus, so some kind of cooling will be necessary. Essentially you're having to get rid of the energy the quasar dumped into the water molecules you're collecting.
OK, the concentration (atoms or kg per volume) of the water, with some back-of-the-envelope calculations. The article states that an amount of water 140 trillion times the mass of Earth's oceans is in the cloud. That's a lot of water! Specifically, since the mass of Earth's oceans is ~1.4 X 10^21 kg, that's ~2 X 10^35 kg of water, about 100,000 times the mass of our sun! But the article also states that it is spread over a region "hundreds of light-years across". If you assume it's 200 light-years across (the minimum to be "hundreds of light-years across"), and assume it's a toroidal region whose volume is about one quarter of the circumscribed sphere, then the volume of that region is about one million cubic light-years, which is 10^45 cubic km or 10^54 cubic meters! That's a lot of space! Spreading that much water over that much space yields an average concentration of ~2 X 10^-19 kg per cubic meter, which is less than one water molecule per cubic cm. This is a hard vacuum! At that concentration, and if you could extract every water molecule from the space you're "mining", to harvest a single kg of water you'd have to mine a region of about 5 X 10^9 cubic kilometers, which is a cube roughly 1700 km on a side. The apparatus to harvest reasonable quantities of water would be gargantuan!
Hmph. Doesn't sound worth it to me. Better to let natural processes concentrate nebular water into things like comets, asteroids, and planets, and then mine those objects.
OK, the water purity. After the big bang there was hydrogen, helium, a little lithium, loads of energy, lots of space (more and more as time progresses!) and precious little of anything else. It takes nucleosynthesis to make significant quantities of the "anything else". Nucleosynthesis occurs in such places as stellar cores, supernovae, and extremely high-energy environments like the accretion disks just outside of a black hole (farther-flung regions of the black hole's accreton disk are too low-energy to support nucleosynthesis). Regardless of the mechanism, in the long run these nucleosynthesis processes never yield one chemical species, like the oxygen to make water. They make all kinds of nuclei: oxygen, sure, but also carbon, nitrogen, iron, cobalt, sulfur, aluminum; various pathways in the various places make essentially all the naturally-occurring elements heavier than lithium.
The cloud contains oxygen to make water, along with the abundant hydrogen that permeates most of our universe. But there will also be nitrogen that with the hydrogen will make ammonia, and sulfur that will make hydrogen sulfide, and carbon that will make methane, and so on. If the observers at the CSO had tuned their receivers to ammonia or methane line frequencies they would have seen those molecules too. But probably weaker: in our sun there are about ten atoms of oxygen for each atom of nitrogen, and carbon is about 5 times more abundant than nitrogen. So while water might be more abundant than everything else...ever taste water with 10% ammonia in it? (Astrophysical measurements suggest lots of stars have carbon abundances higher than our sun's, some double or more!) You'd have to make sure you distill the water out of the miasma, and that takes power.
Net result: it's best to go to the naturally-concentrated sources like comets and asteroids, even if you have interstellar travel.
