There is no consensus on who is right. The punctuation placement, the spelling - editors disagree on all of these points.
First, I should give a note on why the punctuation would change. At the time Shakespeare's plays were published in the First Folio, punctuation had not been standardized as a system for marking syntax. An introductory guide to punctuation in the period would explain that it was flexible; as Walter Ong explained in “Historical Backgrounds of Elizabethan and Jacobean Punctuation Theory.” (PMLA, vol. 59, no. 2, 1944, pp. 349–360), punctuation was partly performative, corresponding to when someone reading aloud might need to take a breath; it was only inconsistently about syntax or elocution:
The present study shows that ... behind the punctuation theory of the time, which was based on a mixed set of principles, there was a factor which is not included under either a syntactical or an elocutionary system as these systems have been understood. This factor was the original theory of the late classical and the medieval grammarians, for whom punctuation was first of all a system demanded by the exigencies of breathing in oral delivery and only secondly (at first rather negatively) a means of interpreting the sense. Lying at the background of Elizabethan and Jacobean theory, this view of punctuation accounts for a practice which is loose by either syntactical or elocutionary standards.
Who set this punctuation? In the case of The Tempest, it could have been the printers of the First Folio in 1623, which was produced after Shakespeare had died in 1616. It could also have been Ralph Crane, a performer in the King's Men, who wrote a manuscript for The Tempest from a possible rough draft. Because punctuation and spelling were not standardized, copiers and printers would often change these elements, and scholars still write about Crane's influence on the text (see Amy Bowles, "Dressing the Text: Ralph Crane’s Scribal Publication of Drama," in The Review of English Studies, vol. 67, no. 280, June 2016, pp. 405–427). So editors would have little qualm about changing punctuation to help readers.
Scholarly editions, recognizing the need to update spelling and punctuation to fit modern reading expectations, and also recognizing that these elements may have multiple valid interpretations, make note of their method of textual editing and, often, choose a primary early text to follow. The Folger digital edition, for example, includes this note after explaining the multiple nature of Shakespearean texts:
The reader of the Folger Shakespeare knows where the text has been altered because editorial interventions are signaled by square brackets ..., half-square brackets ..., or angle brackets .... At any point in the text, you can hover your cursor over a bracket for more information.
In 3.1.15-16, they leave the following note:
But these sweet thoughts do even refresh my labors,
Most [busiest] when I do it. editorial emendation
So they note that they have made an interpretation that goes beyond updates to spelling. Indeed, the First Folio reads:
But these sweet thoughts, doe euen refresh my labours,
Most busie lest, when I doe it.
So the Folger editors, working from the First Folio, have combined what appears to be two words and gotten rid of the punctuation. Here's how they explain the choice in the print edition by the same editors:
This difficult line is printed in the Folio as "Most busie lest, when I doe it, and sometimes appears in editions as "Most busil'est when I do it." There is no agreement about what "busiest" (or "busil'est") modifies, nor about what "it" refers to.
No agreement indeed. Other editors resolve the issue in other ways; this 2002 edition of The Tempest construes the line as
Most busy, least when I do it
with a long paragraph explaining their choice of emphasizing the busy and least rather than trying to make sense of the passage by construing the line as busiest. The comma provides an additional emphasis of their editor's reading, even though it wasn't present in the First Folio.
Who is right? Is inventing one word from two or moving the comma a bigger deal? Those questions approach The Tempest as if an authoritative answer is possible. In this case, we have a manuscript, a First Folio, and subsequent folios, and a line that is difficult to interpret as written. These editors are all trying to make sense of the line, which doesn't quite make sense even to experts. Each edition reflects an effort to balance preserving the text(s) and making sense of the text(s), and in this specific case changes are made based on how the editor reads Ferdinand's previous actions against this specific line. There is no consensus.
