It is popularly claimed that R. Nattan Adler, the mentor of the Hattam Soffer, was indeed of the opinion that the Oral Law may only be written to avoid it being forgotten, and he, who would remember it anyway, was not permitted to write it. (Cf. for example Yalkut Yossef: Pesukei D'zimra V'kriat Sh'ma; notes to chapter 49) This is also mentioned by the K'tav Sofer's son in Hut HaMeshulash (p. 19).
One can certainly question how compelling this argument is, given that writing is a crucial way of conveying information to future generations, but regardless a very similar statement is actually made by his student Hattam Soffer, in a responsum to R. Ts'vi Hirsch Hajes (Vol. I OH #280):
מלבד שהוא עובר איסור דאורייתא דברים שבעל פה אי אתה רשאי לכותבן ולא הותר אלא משום עת לעשות לה' [כגיטין ס' ע"א] ואם איננו עושה לה' הרי איסורו במקומו עומד.
He writes that one who publishes a work with ulterior motives, violates the injunction against recording the Oral Law, which is only permitted for one with the intention of acting for the sake of God.
However, it is important to note, that while he does indeed indicates that the prohibition is still active, and is merely held in abeyance by appropriate exceptions, this statement is made in the somewhat flowery beginning of the responsum, and not as part of an involved halakhic discussion.
Furthermore, it is certainly a minority view. As many have noted (e.g. Hakham Faur S"t in Golden Doves With Silver Dots p. 102), Rambam omits this rule entirely from his halakhic writings, (referencing it only as a historical practice in Moreh Nevokhim I:71) indicating that it is no longer a concern, if it ever was.
The mainstream approach in poskim is to assume as a matter of fact that the prohibition is totally obsolete today, (that is that 'et laasot' totally revoked the prohibition) and it is thus not generally invoked in halakhic works. Like Rambam, the Tur and the Shulhan Arukh, for example, only mention the issue of reciting the Written Law orally, but not the issue of writing the Oral Law. R. Yosef similarly concludes his aforementioned reference to R. Adler by noting that this is not the halakha.
This is particularly understandable if the whole prohibition is rabbinic as the Yereim (268) implies (noting that the prohibition to recite oral Torah is rabbinic, and the cited verse which is the source for both laws is a mere asmakhta. If it was the rabbis in the first place who forbade it, they have the licence to fully revoke it if they found it to have become counter-productive.
However, it should be noted that R. Qafih writes (Ketavim Vol II p. 546-7) that the prohibition of writing the Oral Law is still active, and therefore Rambam only recorded already recorded laws in the Mishneh Torah.[i] [ii]
Nevertheless, an important corollary of R. Qafih's assertion, is that explicating already recorded oral law is not in violation of the injunction. Although one can question where the line is drawn between deriving information from already recorded texts, and recording hitherto unrecorded information, this perspective would likely render the issue moot, as unlike in the time of the Ammoraim when much ancient oral teachings were still promulgated, most modern literary activity focuses on explaining existing texts, rather than recording ancient otherwise unrecorded traditions. Even were there such traditions, it would likely be permissible to record them under the rubric of "et laasot". Thus, if it is permissible to record and explicate already recorded oral law, and it is permissible to record otherwise unrecorded traditions, then little is left forbidden.
[i] He noted that cases where Rambam says "yireh li" are still based off already recorded sources, but the sources are cryptic.
[ii] While this is not the place to discuss this view in depth, it should be be noted that it seems difficult to maintain that everything in the Mishneh Torah is based on explicit written sources, rather that oral tradition, given for example Hilkhot Avodah Zara (12:6). Perhaps R. Qafih would answer that that tradition itself can be derived from existing sources, cf. Tashbets (2:100).