If the House don't appropriate it the Senate can't spend it. The House has all the power here, they just fear being accuse of shutting the government down.
The Senate confirms Presidential nominees. With the elimination of the filibuster for nominees and todays SCOTUS ruling getting a majority is critically important.
Additionally nothing from Congress makes it to the Presidents desk without a Senate vote. The Dems have been blocking votes on legislation to protect red and purple state Dems from voting on uncomfortable legislation AND to prevent Obama from having to veto popular legislation.