In order for a liquid airlock to work some points must be considered. They have been mentioned in some of the other answers, but I will try to combine them and suggest some new implementations.

 - Gravity.

    As stated there must be some gravity present for this to work in principle. The required column height (see sketch in the question) or pressure head is h = p/(rho*g). To compensate 1 atm habitat pressure on the moon a water column would need to be about 60m high, and a mercury column 4.5m. On earth 1 atm corresponds to only 760mmHg.

 - Vapour pressure of the liquid vs. pressure of the atmosphere (if any).
   
   The rate of evaporation depends on the external pressure as well as the intrinsic vapour pressure of the liquid.
   Liquid metals have low vapour pressures, hence they where suggested in the original article. Water on the other hand has a much higher vapour pressure and will evaporate/ freeze and sublimate very quickly in space or on a body without a significant atmosphere, e.g. the moon. There are some other possibility for liquids from this stand point. Silicone oils for example are used in vacuum applications because they can have low vapour pressure. Quite recently (compared to the article's publication date) room temperature ionic liquids have been studied which have very low vapour pressures and have also been used in vacuum applications, e.g. in [this article][1].

   In any case there will always be a small rate of evaporation on a body without any atmosphere, which means you should top up your air lock once in a while.

 - Practicalities of passing through the liquid airlock like buoyancy forces and reactivity of the liquid.

   Liquid metals like Metals like Mercury and Gallium tend to be very reactive and will amalgamate (react) with most other metals. NaK is very corrosive as well. This presumably be solved by coating exposed portions of cargo with inert polymers like PTFE. Silicone oils on the other hand are very inert and should pose no problem at all.
   David Richerby mentioned that also buoyancy can be a problem, especially with the high density of metals. Since buoyancy force is independent on gravity (see [xkcd's What If][2]) diving through roughly 5m of mercury in your non metallic space suit on the moon  would be next to impossible unless you had big Tungsten or Platinum ankle weights.


So to sum up it would be pretty difficult to make as well as use a liquid airlock. The volumes and masses of the liquid alone would be way to much to carry even to the moon. Maybe you could extract some of the resources in situ, however to me it seems this type of airlock is totally impractical unless you already have a well developed industry on another planet. Although it is a nice Physics and Chemistry problem to think about.

  [1]: http://cdn.intechopen.com/pdfs-wm/40002.pdf "Ionic liquids in a vacuum chamber"
  [2]: https://what-if.xkcd.com/124/ "Lunar swimming"