The blame game is probably pointless if you are thinking it will lead to electoral gains for either party in the 2018 elections. It's unlikely to matter in the end, unless this shutdown lasts a really long time.
It's important to examine the root of the problem, though, since it has basically broken our government with these shutdowns and constant shutdown/debt default threats since at least 2011. The Republicans have a majority in the House, but it's a majority that is basically a coalition that relies on the House Freedom Caucus. If the Speaker crosses them and passes any bipartisan legislation with the Dems, the consensus seems to be he'll lose the House majority who will support him remaining as Speaker. This is why Boehner made a deal and passed bipartisan legislation, with the promise to the HFC he'd resign. He basically fell on his sword.
Then there's the issue of the people that Trump has surrounded himself with, who have contributed to sabotaging any DACA deal. WH Advisor Stephen Miller: garbage white nationalist. WH Chief of Staff John Kelly: probably the same, based on reports. Kelly probably already some white nationalist leanings before, but his time as the head of DHS probably exacerbated them. Tom Cotton: You never want to be taking the advice of this person. Mick Mulvaney: Huge mistake appointing a Tea Party guy who hates the federal government as the head of the OMB.
And please, let's stop treating DACA and CHIP as some sort of 'gift' to Democrats that should be used as a bargaining chip. These things have widespread support. Democrats would be making a mistake if they are willing to sell the farm to get things that so many people in the electorate (and allegedly Congress) support.