Water desalination is already practiced on industrial scales. While obtaining water via desalination is virtually always more expensive than taking in and processing fresh-water directly (i.e. from lakes, streams, and underground reservoirs), in many places (e.g. parts of the Middle East) with scarce natural water reserves, entire population centers have become dependent on desalination plants for their water supply.
Desalination plants usually use either distillation or a process called reverse osmosis. Distillation is the simpler technology to understand, basically you simply pump in salt-water, you then heat it up (usually at reduced pressure), and finally you collect the steam. Usually this process is done in several stages (e.g. as roughly shown here) simply because doing so is more energy and cost effective than simply distilling the water in one step. Most of the water processed via desalination is obtained using some distillation based scheme.
The second major approach, reverse osmosis, is in principle similar to the filtration scheme you proposed. The way reverse osmosis works in practice is that you have a membrane with very small pores (on the order of Angstroms), such that water molecules can pass through, but not charged ions. The difficulty of purifying water in this fashion (in contrast to normal filtration) is that it must overcome the osmotic pressure of the system. Briefly, water molecules have a thermodynamic driving force to flow from an area of lower solute concentration to areas of higher solute concentration, creating a pressure called the osmotic pressure. However when carrying out desalination you want to reverse this process (hence the name) in order to pass pure water across the membrane. This can achieved by applying large pressures to the salt-water (upwards of a few atmospheres) in order to overcome the osmotic pressure and push the water through the membrane. The total process is illustrated graphically here.
There are certain environment concerns posed by desalination plants, such as damage to the local environment posed by rapid water intake (e.g. sucking in fish and other marine organisms), water pollution caused by releasing the byproducts (a combination of brine and various processing chemicals), as well as the total energy usage needed to power these plants and the challenges that poses down the line. However, running out of water is not really a major concern, at least as far as the oceans are concerned. It's not just that the total amount of water stored in the oceans is incredibly vast to begin with, but more importantly virtually all of the water that is desalinated will ultimately be evaporated and find its way back into some kind of a reservoir. By desalinating the water, we are not changing it molecularly, so the total amount of water on Earth does not appreciably change, we just shuffling it from one place to another.
