I don't think so, mainly because DNA isn't a language. It's convenient to think of it as one, but it's just an analogy, and it breaks down when you look closely. The properties of DNA are a function of the physical makeup and structure of the molecule itself. It's what gives it its shape, what allows proteins to bind to it, and what allows it to be translated into RNA and then into proteins.
It's closer to the workings of a machine than it is to a language. As such, I thin a base-anything system would be too simplistic to describe it. It would be like trying to reduce a car engine into a numeral system... and we're just talking about a few bases at a time, here. An entire gene is much more complex, so it's more like a car engine made out of car engines. OK, my analogy is starting to break down, but my point still holds.
So we can describe DNA using a base-4 system, in a sense, but there's not much you can do with it. Some things behave in a consistent manner (like the 3-base codons that code for specific amino acids), but there's a limit to what we can tell by simply knowing the sequence. We can't know what it's going to do until we actually observe it in action.