Rather than trying to address the claim of who is to blame, I will give you timeline of events and let you decide for yourself who deserves how much of the blame.

19th December, 2018:

Senate passes with a vote of 100-0 a bi-partisan short-term spending bill without funding for Trump's wall. Bill is expected to pass the House and be signed by the President. [1]

Fox and Friends, Rush Limbaugh, and Ann Coulter criticize Trump for "folding" on the wall. [2]

20th December, 2018:

The president informed us that he will not sign the bill that came up from the Senate last evening because of his legitimate concerns for border security.

-- Speaker of the House Paul Ryan (R)

Instead of voting on the bill the Senate passed, the House with Paul Ryan (R) as Speaker passes a spending bill with $5 billion in border wall funding. Bill is not expected to pass the Senate, and ultimately did fail in the Senate, where 60 votes were needed and Republicans only had 51 seats. [3] [4]

3rd January, 2019:

The new House of Representatives with Nancy Pelosi as Speaker (Democrat) passes a bill mirroring the one that passed the Senate. Senate Majority Leader Mitch McConnell (R) blocks the bill in the Senate, saying he will not bring a bill to vote without the president's approval. [5]

NOTE: Please remember comments are for asking for more information or suggesting improvements. They are not for complaining about answers, or claiming that accepted answers somehow don't answer the question.

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.

Passing a spending bill in the Senate requires bi-partisan cooperation because it requires overcoming a 60-vote requirement. The President demanded that any spending bill include a line for $5 billion (about 0.13% of the total yearly spending) for a border barrier (which he colloquially called a "wall").

The Republicans were willing to include the line for this spending. The Democrats were not. By this logic, the Democrats have a partial responsibility for the bill not passing.

Clearly they are not the only ones to blame. The President's unwillingness to give up on funding of "the wall" is to blame as well. But using the cloture rules to hold up a spending bill over a 0.13% spending line is the reason (or logic) for why the President is putting the failure to pass the spending bill at the Democrats' feet.

This isn't really that difficult: It's a typical standoff in which neither side wants to budge. Democrats could vote for the wall, but haven't. So, sure, it can be considered at least partially their fault. That's not to say Trump isn't to blame, either. But, if ending the shutdown were enough of a priority, then Democrats could simply vote for the money for the wall, and be done with it. If it's not, the standoff will continue.

If Republicans have a majority in the House and the Senate, and also have control over the executive branch, how could it be possible that the Democrats are responsible for the shutdown?

Answer: The Republicans' majority in the Senate isn't large enough.

In order to end debating a bill, 60 senators have to agree to start voting. Because the Republicans only held either 51 (just before the 2018 election) or 53 (after the 2018 election), the Democrats have enough votes to continue debating certain bills forever. This is known as 'filibusting' a bill.

As a result, the Republicans were not able to pass a bill that included the requested money for a border wall, even when they had majorities in both chambers of Congress.