It is hard to say that the shutdown is due to one particular party only, despite the claims made by Democrats and Republicans alike. The truth is that either party has the choice to concede and end the shutdown, but is choosing not to, and hence neither party is innocent, both are using the prolongment of the shutdown as a 'threat' to get what they want.
However, that said, Donald Trump took the blame of the shutdown on live television, claiming multiple times that he would "own it".
Why does Donald Trump willingly take the blame for a monstrously unpopular thing that pretty much everybody wants to see end?