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Yale has changed their language, but I don't see a clear alternative here.
At this point, I think referring to students by their specific class years ( "First-Years and Sophomores" vs. "Juniors and Seniors") is the best alternative for now, particularly because of the ambiguity around what class years the terms "underclassman" and "upperclassman" really refer to.
Referring to years verbosely makes sense to me.
First-years and sophomores
First and second years
Or if you want a different meaning
First through third years
The phrases "lower-division students" and "upper-division students" are often used to refer to students in the first two years or the last two years (respectively) of a four-year degree. See, for example, this page from U. Texas Austin:
Undergraduate students are classified as freshmen, sophomores, juniors, or seniors, based on the number of semester credit hours passed and transferred, regardless of the hours’ applicability toward a degree. ... Freshmen and sophomores are referred to as lower-division students; juniors and seniors, as upper-division students.
See also this page and this page from Cal State classifying transfer students in this way. This dichotomy is also (and perhaps more frequently) applied to courses, for examples, see UCLA and U. Washington.
I think Yales' terminology is appropriate, though I find it may be more prudent to refer to the program being attempted (undergraduate, graduate, master's, doctoral, post-doctoral, etc) and just append First-Year; is it important to dictate which year they are in aside from the first?
If more specificity is truly needed, Last(Final?)-Year may also be applied in that way, or 'Last-Year Graduate Student.' If in the middle years, state the year as suggested. That seems more than appropriate.
First/Last go together, as opposed to Initial/Final, so First/Last would be at least consistent in its lexicon.
Such as a First-Year Undergraduate being a literal first of the first year.
A First-Year Graduate student being the first year of the graduate program. Etc.
My own criticism of my suggestion is that it may not translate into other academic systems cleanly if their progression structure is different. I can only speak for the American way.
I cannot yet comment so I apologize for responding to the other answer and if inappropriate can be edited out by whomever. In regards to 'themself', it does give me a bit of trouble in syntax. Despite being in both communities it is relevant (linguistics and LGBT), it feels wrong to say out loud. In my own writing and having to reference, speak with and about gender-neutral individuals, 'themselves' is read just as well and can be the same as 'themself' depending on the syntax of the sentence. "They completed the task themselves." as long as the article is defined already, it sounds appropriate. If it isn't, it will sound, again, a bit strange. "Ash completed the task alone." resolves the issue completely, and avoids repetitive pronoun usage.
This is a failure of English with the lack of a third person singular neuter. Many other languages have it. Apologies again for the digression, but it may still be valuable in gender-neutral academic writing. I would appreciate any comments or perpsective on this.
I doubt that you will find a widely used term at the moment. Language takes a while to catch up to social changes and the standards of what is acceptable differ around the world. Most new changes to the language are seen as clumsy at first and take a while to reach acceptance and it may initially be only locally in any case. Even this system, which tries to be gender neutral, balks at "themself".
Change will come. Do the best you can in the short term and be aware that some choices are exclusionary.
And even "underclass" has a social meaning outside academia, that may be unintended. Language is hard and only imperfectly mapped to thought processes.
