How would a DFA be designed to accept a language defined as the string comprised of every other character in each string in another language?
This is a closure property. For language $L\subseteq \Sigma^*$, define the language $even(L) = \{ x_2x_4\dots x_{2n} \mid x_1x_2x_3\dots x_{2n-1}x_{2n} \in L, x_i\in\Sigma\}$. Now for regular $L$ the language $even(L)$ is again regular. This is a special case of closure of the regular languages under finite state transductions.

In case of this specific property an automaton $M_1$ for $even(L)$ when $M$ for $L$ is given, can be found as follows. For every pair of consecutive transitions $(p,a,q)(q,b,r)$ for $M$ introduce the transition $(p,b,r)$ in $M_1$.

Additionally one needs a somewhat formal argument that this will in fact work. 

Alternatively, this closure property can be deduced from known closure properties: (inverse) morphisms and intersection with regular languages.
Make a copy of $\Sigma$: $\bar\Sigma=\{\bar a \mid a\in \Sigma\}$.
Define the morphisms $g,h:(\Sigma\cup\bar\Sigma)^*\to \Sigma^*$ where $g$ removes the bar (but keeping the letters), and $h$ removes letters from $\bar\Sigma$. $R$ is the regular language $(\bar\Sigma\Sigma)^*$
Then $even(L)$ is the composition $h(g^{-1}(L) \cap R) $.