This is going to be anecdotal, but I would say at this point it's impossible to draw a conclusion.
As far as I know
Heinlein never said if the Mobile Infantry is based on any specific branch.
Heinlein's military service in the Navy would push me towards the Marine Corps as the influence.
The quote "Come on you Apes, do you want to live forever?"
is generally attributed (it's possible other versions were uttered before) to Dan Daly, a Marine. Roughnecks, is a play on Leatherneck, a moniker for Marines.
Even their employment within the story, flying along with "the Fleet" until being inserted at their destination is very Marine like.
But without the Roughneck pun and the Dan Daly
quote, I could draw conclusions that would face me towards the Army.
Starship Troopers is on the Commandant's Reading List, also on the Navy Recommended Reading List as well.
My opinion is skewed as I am a Marine, but after reading the book and seeing the movie, the portrayal always seemed Marine Corps biased.
That would be more familiar to Heinlein considering his service in the Navy.
scifi.stackexchange.com/users/12929/McFuu
I'd note the comparison he makes in passing - that an MI is selective in a way that orbital bombardment simply isn't - implies that he considers both to be weapons deployed by the Navy, as opposed to merely transported by them on occasion.
That fits the Marines much better.
Further, the lack of armor support makes it harder to reconcile the MIs with the GIs.
(Admittedly, the point of the MI suit is to be armor support, but it feels more like an upscaling of elite infantry rather than a downscaling of tanks or choppers.)
A rapid, well-armed unit supported only by naval gunship batteries?
Sounds like Marines to me.
scifi.stackexchange.com/users/8191/rsegal
One thing that leans more to a naval/marine flavor would be when Rico was going through his officer candidate school.
Right before shipping off for his tour as a "temporary, probationary and supernumerary" third lieutenant, Rico and several cadets were having a conversation with Colonel Nielssan.
Many of the stories of the pips each cadet would wear revolved around naval actions or combat involving officer’s assigned to the fleet.
Additionally, ranks, customs and spoken verbiage all decidedly revolve around a naval theme.
While a modern airborne ranger battalion would be assigned some of the raiding/take and hold missions as an M.I. unit does, this book was written in the late 50's, much earlier the modern concept of heli-born assaults.
Taken as a whole, I would say that Heinlein leaned more towards a marine styled always deployed always ready force than an army styled airborne unit.
scifi.stackexchange.com/users/4289/Chris
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.
scifi.stackexchange.com/users/2834/KeithS
The Mobile Infantry concept is based on Heinlein's own experience as a Naval officer and contemporary military doctrine.
At that time, military leaders were experimenting with the 'pentomic' army.
They were concerned with survivability on the nuclear battlefield.
The most notable aspect of the pentomic system was the five part manning and C2 system vs the more flexible and successful three part system, five squads per platoon, five platoons per company, etc.
Combine this with his interest in space travel and a good imagination
he came up with the Mobile Infantry.
Rocket ships replace Dakota's; atomic rockets replace bazookas; flamethrowers are still useful.
The infantry still dies.
Interestingly, the US DoD is actively pursuing many MI concepts, most notably, the mobility suits used by the infantryman in the 23rd Century.
"Starship Troopers" is an excellent book on leadership as much as it is a future history.
It is required reading in many military schools.
scifi.stackexchange.com/users/77085/Trajan Aurelius
A comment on  this question  got me thinking - was Heinlein explicitly modeling Mobile Infantry on US Marines? Or More the Army?  I'll accept either a word-of-god answer (e.g. sourced from Heinlein himself), or a good analysis showing how specific details of MI are patterned more on the Army or the Marines of the time.  The question specifically pertains to the Heinlein book, not the movie that accidentally shares the same name with it.
Was Mobile Infantry in Starship Troopers modeled on the Marines or the Army?
scifi.stackexchange.com
<robert-a-heinlein><starship-troopers>
scifi.stackexchange.com/questions/32843
scifi.stackexchange.com/users/976/DVK-on-Ahch-To
Heinlein never said if the Mobile Infantry is based on any specific branch. However, the portrayal seemed Marine Corps biased. Although it is impossible to draw a form conclusion, the Starship Troopers is modelled more on the Marines.
Although it is impossible to draw a form conclusion, the Starship Troopers is modelled more on the Marines.