computers work in different ways to biological systems. A biological system translates the genetic information to build proteins - it works like a mechanical machine.
Computers on the other hand are logical machines, and although there is no logical reason why computers cannot use the digits 0-3, there are definite practical reasons.
A particular part of a computer can only have two states - on or off. Whether this is a bit, a bus or whatever, it can only have two possible states. If you wanted to use a quaternary system then you would need to have found a way for the computer to represent these different possible states of a bit, connection, etc.
A simple way to do this would be to use two bits to hold four possible values. But this is overly complicated and it would be much simpler to just use two states.
As short answer, computing and DNA are two very different systems which are too incompatible to compare. Computers simply cannot practically have more than two states. Rather they use bytes (groups of 8 bits) allowing for numbers from 0 to 255. Likewise DNA uses coding triplets which (4 * 4 * 4) which allows for a maximum value of 64 different values.
hope this helps
raina_vissora said that DNA is binary. This is not strictly true as although AT, CG are the only possible combinations, the order which they are held can be reversed. EG.
ACG TTA GTG are all acceptable triplets.
DNA is not read 'by the rung' it is only read only on the coding side (apart from transcription) , meaning that there ARE 4 different values possible as a sing 'bit'