Honestly they should have caved a long time ago. Trump and his cronies are loving the shutdown.
No, they absolutely should not have, and should continue not to. The message they should be receiving from the most recent elections is that the Republicans are losing support, not gaining it.
What they should be doing is the opposite. It is in the GOP's best interest to reopen the government. Until they can see that their actions are losing them all of their support, ask for more.
For every additional day that someone starves due to the government's failure to provide, ask for something to compensate them. Every additional day that people work without pay, demand something in return for them.
Eventually they will need to cave and compromise. Move the bar so far back that they have no choice but to compromise harder.