The right choice depend on what are your needs.
Liquid layouts are perhaps more difficult to understand and implement, but you gain flexibility.
A flexibile layout means it can easily adapt to different screen resolutions.
Once you have your liquid layout done, you can possibly adapt it also for mobile websites (if it isn't already) with a couple of changes, mainly CSS rules.
With the fast pace at which new technologies come out (new devices, new screens, new resolutions, 16:9, 16:10, Full HD etc), you can easily implement a layout which has no problem displaying at various resolutions on different displays.
This is what liquid layouts are for.
On the other hand, a static website is nothing more than what we've seen in the past years, like a centered 960px wide page, unable to being displayed on most mobiles, and even if they display it, good luck navigating the website.
Note that you can actually implement a sort of hybrid: a fixed size layout, with a max width of let's say 960px to be viewable in desktop browsers normally, which then adapt to lower resolution with an auto width.