Contrasting some other answers, I feel that the sheer number of MI, the lack of another separate force of ground troops, and their basic status as the "regulars" of Earth's military, generally lends itself to an Army-style designation for the Mobile Infantry. Their landing-craft-style insertion is vaguely Marine-ish, but they're space-faring soldiers; other than space-jumping Airborne-style (hinted at in the book, absent in the movie) I can't think of another viable way to get boots on the ground. In space, there would be no significant distinction between air and sea as there is with the modern military, and so being ferried into combat via space landing craft is roughly analagous to being delivered either by C-130 or by LCAC.
The modern Marine Corps is a semi-self-sufficient "first response" force that can field comprehensive military power quickly, and as such are composed of land, air and sea vehicles, designed and chosen with an eye for portability and versatility. The amphibious assault ships they commonly use, of the Tarawa, Wasp and brand-new America classes, are technically under the control of the Navy, but their sole purpose is to get Marine Expeditionary Forces into the theater.
In terms of manpower, though, the USMC is the smallest of the US Armed Forces branches, roughly one-fifth the total size of the U.S. Army (not counting the Army's civilian manpower component, which is about the size of the USMC just by itself). The modern Army is well-used to being carried into battle however they can get there; in WWII, Army troops and their materiel were delivered by troopship, Duck landing craft, C-47 paratroop plane, and even glider. Once they land, they're pretty much walking, much like we saw in ST.
However, why does it have to be one or the other? There may be one "Mobile Infantry", but there are various units within it, some of which may be better suited for "hopping" from place to place instead of maintaining a more static "front line" like an army is normally thought to do. All the branches have elite special forces, and traditional units of same: Army Rangers, Navy Seals, Marine Force Recon, etc. All of the branches now contribute complements to SOCOM, which, taken as a single unit of "Special Operations", could very readily be replicated within MI.