The House party structure doesn't apply.
Article II, Section 2 of the Constitution provides that the President has the “Power, by and with the Advice and Consent of the Senate, to make Treaties, provided two thirds of the Senators present concur.” Ratification is thereby approved or rejected by the Senate.
> As of 2013, the majority of United States free trade agreements were implemented as congressional-executive agreements. Unlike treaties, such agreements require a majority of the House and Senate to pass. Under trade promotion authority (TPA), established by the Trade Act of 1974 and renewed by the Trade Act of 2002, Congress authorizes the President to negotiate "free trade agreements ... if they are approved by both houses in a bill enacted into public law and other statutory conditions are met.
In the case of the USMCA, the Trump administration is seeking approval under "fast-track authority", which he has under the Bipartisan Congressional Trade Priorities and Accountability Act of 2015. (This authority was originally set to expire in 2018 but was successfully extended to 2021.) [2]
The effect of fast-track authority is that when "implementing legislation" for a trade agreement is considered by Congress, no amendments can be proposed, debate time is limited, and (as a result of time being limited) the bill cannot be filibustered in the Senate. This makes it much easier to pass.
However, the process only applies to legislation, i.e. to the congressional part of congressional-executive agreements. Passing an agreement as a true treaty would still have to follow the normal Constitutional procedure requiring a two-thirds majority in the Senate, which is generally considered more difficult than getting a bare majority of both houses. I'd say that applies even in the current political situation, because the Republicans will only have 53 seats in the Senate; a two-thirds majority is 67 votes, so they would need 14 out of 47 Democrats (30%) to vote yes, assuming all Republicans voted yes. In contrast, in the House the Republicans will have 200 seats, and would require 18 out of 235 Democrats (8%). On the other hand, the House rate would face an opposing majority leader, and the House can also vote to withdraw fast track authority. [3]
In any case, regardless of what the administration could theoretically do, it's currently trying to pass the USMCA as a congressional-executive agreement, which means the change in House party structure certainly is significant.
By the way, even if a true treaty is passed, in theory the House can refuse to provide funds to implement it, which is arguably[4] allowed under the Constitution. But admittedly that's very unlikely to happen unless they really hate the treaty, which wouldn't be the case with the USMCA.
[1] https://en.wikipedia.org/wiki/Fast_track_(trade)#TPP_and_201...
[2] https://www.lexology.com/library/detail.aspx?g=5b4c71fc-b979...
[3] https://fas.org/sgp/crs/misc/R43491.pdf (page 27)
[4] https://www.govinfo.gov/content/pkg/CPRT-106SPRT66922/html/C...