Republicans control both houses of Congress and the White House. A government shutdown and its consequences would fall on their shoulders.
Thanks to this deal Democrats have ALL the leverage here. They shouldn't fold. If there is a shutdown, it won't last long before the Republicans come crawling back to the table with a raft of concessions.
It isn't just a shutdown at stake, it is also the debt ceiling. If the US government defaults on its debt, it could literally destroy the world economy.