Can you say anything interesting about a language knowing only that it is prefix-closed?
If $P$ and $Q$ are languages closed under prefix, then
$P\cup Q$, $P\cap Q$, $P \cdot Q$, $P/Q$ are also closed under prefix. Also you can think of such a language $L$ as a directed tree. Define the graph/tree $G = (V,E)$ where $V = L$ and
$$E = \{(\mathit{prefix}(w),w) \mid w\in L \}$$
(the words are the nodes and the parent of every word is its prefix).
Note that I assume that $\mathit{prefix}(x) = \varepsilon$ for all $x\in\Sigma$, meaning that the prefix of a single symbol is the empty word. Thus the root of the tree constructed above is the empty word. The constructed tree can be easily transformed to a DFA recognizing the language (I think you can see how).