I’m sure there is something that I am missing (certain vote percentages, loopholes, who knows), but to me the logic seems that if Republicans had the ability to pass the funding measure and then didn’t, wouldn’t the shutdown be the Republicans fault?
I think there are a couple things: first, the new Democratic majority; and second, the filibuster/cloture process in the Senate.
Yesterday (3 Jan.) was the first day of the 116th Congress. As of yesterday, the Democrats have a majority in the House (235 to 199, one disputed seat) and the Republicans have a majority in the Senate (53 to 47). At this point and going forward, both parties bear responsibility for passing or failing to pass spending bills.
However, before yesterday, the Republicans had a 236 to 196 majority in the House (three vacant seats) and a 51 to 49 majority in the Senate. In December, the (Republican) House passed a spending bill with funding for the president's proposed border wall. The Senate unanimously passed an alternative short-term spending measure without border wall funding, which the president then threatened to veto. Following that, the Senate Majority Leader stated that he would not support (or presumably schedule a vote for) any bill that the president threatened to veto.
Also, almost all bills in the Senate require 60 senators to invoke "cloture" in order to end debate and vote. Bills that fail to receive cloture are "filibustered," and given that Senate Republicans had an extremely slim majority in the last Congress, invoking cloture against a united Democratic conference was quite tough. Even in the new Congress, invoking cloture will be tricky for polarizing legislation (e.g. anything dealing with "the wall"), albeit marginally easier for the Republicans than in the last Congress.
In my opinion, anyone who assigns blame or responsibility for the shutdown to one party exclusively is trying to spin the facts to fit a partisan or ideological narrative. How you assign blame depends on your personal beliefs, what you think about the majorities in Congress and what you think about the filibuster.