To be fair, the administration has made several concessions (steel instead of concrete barrier, a multi step funding proposal, DACA extension, etc), while the Democratic leadership has been unwilling to compromise on increased funding for border security. It is on both sides that this shutdown has gone on so long.
The shutdown began under a government where all 3 branches were under Republican control when the Admin. refused a bill that had already cleared both houses, the Senate via unanimous consensus. So the shutdown is absolutely the direct result of the Admin. In regards to its continuation, were the Democrats to yield this would set a precedent that a shutdown can be used to force negotiation at gunpoint with real people used as hostages.
I really don't see how this is anything except an attempt for the Admin. to save face over a failed campaign promise to build a wall that would be paid for by another nation. The senate Republicans refusal to approve the same bill they unanimously voted for is the reason the shutdown continues.