The baseline obligation is broad and demanding: Menachot 44a records Rav Sheshet's ruling that one who does not don tefillin violates eight positive commandments, and Yoma 33 establishes that one may not skip even the arm-tefillin in order to reach the head-tefillin more quickly, so pressing is the duty — a stringency that רבינו בחיי, כד הקמח תפילין reinforces by teaching that every Jew who is healthy and physically capable is fit to wear tefillin, without needing a degree of bodily purity as exceptional as that of Elisha Man of Wings.
That obligation extends fully into travel: Menachot 36a records that one who rises early to travel and fears his tefillin may be lost may don them even before the proper time, touching them and blessing when the time arrives, and Contemporary Halakhic Problems, Vol III, Part I, Chapter II Travelers notes that Jewish law provides extremely few concessions or exceptions for the traveler, whose obligations are frequently rendered more difficult, not lighter, by changes of time and locale.
The primary exemption relevant to aerial travel concerns distress and an unsettled mind: Shulchan Arukh, Orach Chayim 38:9 rules that one who is distressed or whose mind is not settled and ready is exempt, since it is forbidden to take one's mind off tefillin once they are worn — a ruling echoed in Mishneh Torah, Tefillin, Mezuzah and the Torah Scroll 4:1–14, which adds that bodily cleanliness is equally essential, so that anyone unable to maintain it is similarly exempt, as illustrated concretely in Chullin 110 where Rami bar Tamrei is excused on account of intestinal illness.
Nighttime removes the obligation entirely: the Mekhilta reasons from a comparison with mezuzah that Mekhilta DeRabbi Shimon Ben Yochai 13:8 the word 'from days' comes to exclude nights, and Menachot 36b records the Talmudic debate about twilight and Shabbat that presupposes tefillin are in principle a daytime mitzva, so a balloon flight conducted entirely at night would not carry the obligation at all.
Mishneh Halachos (Mishneh Halachos 11:77) addresses the laws of prayer during airplane travel, ruling that the traveler's prayer should be recited immediately upon the aircraft lifting off, which indicates that halachic obligations — including those tied to transit — take effect the moment one is airborne, a framework directly applicable to balloon flight as well.
B'tzel HaChochma (B'tzel HaChochma 2:16) rules that one who sees seas, mountains, or rivers clearly from a great altitude, including from an airplane, is obligated to recite the appropriate blessing over them, establishing that aerial observation of the ocean from above triggers halachic duties — a principle that supports treating the balloon traveler over the ocean as halachically present and fully subject to applicable obligations rather than suspended from them by virtue of his position in the air.