If they do establish a protectorate or dominion or whatever they decide to call it despite the fact that they don't have much to gain (Imperialism for Imperialism's sake wasn't exactly an unheard-of mentality at the time) I could easily imagine the US becoming a Central Power in WWI as Britain can gain a fair amount of food from Latin America ITTL but on the flip side has a half-century of enmity built up between them and the USA, which coupled with the latter's high Irish and German populations could result in a push towards the Germans and their allies in the First World War.
OTL, Argentina was known as Britain's bread basket. in WW1, virtually all trade was conducted on the part of Britain/Allies due to the blockade. The situation doesn't change because Britain takes more of a hands on role in Argentinian Gov't.
Don't see the Irish/German situation changing in the US. Argentina did attract a sizable Irish immigration, OTL, so maybe more head to US and away from British protectorate?
OTL, South America was recognized as a primarily British Sphere of Interest (US didn't make much inroads til after WW1), so I don't really see where the enmity between Britain/US is radically changing. Butterfly wings flapping might alter the world situation, though. I have a hard time seeing US being driven away from a very nice partnership with Britain, at least not enough to actively opposing them.
IF they did, though, and we see a French influenced Mexico and a British influenced Argentina, and everything else stays the same (logically things are changing simply to allow those two things to change, but such changes enter so many variables we won't recognize the equation) , and the US sides with Germany... interesting. US militarily isn't going to make a huge difference - Britain/France still rule the seas, but US might make enough difference to interrupt shipping enough to starve Britain into submission. And US manufacturing, while probably still mostly blocked from reaching Germany, won't be going to France/Britain (Russia, OTL was walled off by Turkey). IF mexico sides with France, USA has a hot border to protect, so US troops are likely to stay in North America, or lost to the bottom of the sea trying to get to Europe.