The main reason is that binary logic is so much more effective than logic using more than level at the transistor circuit level.
There are alternative number systems that are in use but those are implementations of different kinds of numbers or numbers with different internal structure.
Such as the redundant number system which have parallel addition and multiplication that does not have long serial paths that the normal operations have. This means it does addition in less than log n time.
The ternary numbers are functionally equivalent to binary only differing in representation but not any other aspect so they provide no advantage.
