Given regular language $L$, is $L_1 = \{ w \mid \text{each prefix of } w \text{ of odd length} \in L \}$ regular?
Try constructing an NFA for $\hat L:=\{w|$ there is an odd length prefix of $w$ that is $\notin L\}$, given a DFA for $\overline L$, and then show $L_1=\overline {\hat L} $.

If I'm not mistaken, this should work out.