Note that in algebra one usually writes $H^{-n}$ instead of $H_n,$ so connective and coconnective algebras live in the same bigger category of (cohomologically graded) DGA's. (There is also a topological convention of writing $\pi_{-n}$ instead of $H^n$). If $A$ is connective is has cohomology in negative degrees, and there is a map $A\to H^0(A).$ If $A$ is coconnective, then there is a map $H^0(A)\to A.$ 

Now $K$ theory is properly an invariant of a(n $\infty$-) category, not an algebra. It is never interesting to take the category of all $A$-modules (whose $K$ theory is trivial by the Eilenberg swindle), so you always want to impose some finiteness condition. Usually the $K$ theory of $A$ is defined as the $K$ theory of the category of perfect modules. In this case, given a map $f:A\to B$ of DGA's, the functor of restriction $B\mathrm{mod}\to A\mathrm{mod}$ does not (in general) preserve perfect objects, so the only natural map on $K$ theory is the one associated to the induction functor, $M\mapsto M\otimes_A B$ from $A$-modules to $B$-modules. Induction from $H^0(A)\to A$ in the coconnective case does not induce an isomorphism on $K$ theory in the very simplest nontrivial coconnective dga, $k\oplus k[1]$ (square zero extention), where the induced map on $K^0$ is trivial (any perfect module over $k$ has equal total dimension of even and odd cohomology). In the connective case, you can also cook up a counterexample (though this is more tricky). The paper linked in Maxime Ramzi's answer gives a very restrictive condition on coconnective $A$ which guarantees that the map is indeed an equivalence on $K$ theory. 

However there are other categories of modules that one can study. If $A$ has finite-dimensional total cohomology, it is interesting to study the $K$ theory of the category of all $A$-modules with finite-dimensional total cohomology (which is bigger than the category of perfect objects); more generally one can also look at the related category of modules whose cohomology is finitely generated as an $H^0(A)$ module (a generalization of "coherent $K$ theory" of an affine scheme). In both of these cases, every $A$-module has a filtration (the cohomological filtration) whose associated graded subquotients are concentrated in a single cohomological degree. This means that pullback along the map $H^0(A)\to A$ is a valid functor which induces an isomorphism on $K^0$ (and, at least in the coconnective case, on higher $K$ groups as well).