We have a regular set and we want to remove all strings such that in our set for any string, we should not have its prefix. Now consider the DFA for $L$. Take all of its final states and see if there is a path to any other final state. (This is a graph search problem). If so, then mark that final state as non-final because this path we found is essentially adding a "suffix" to an already accepted string in $L$ and making another string in $L$. Doing like this for all final states give us the DFA for $\text{MIN(L)}$. So, it must also be regular.