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Presumably the ships of Axum were similar to the earlier ships of Egypt on the Red Sea. The ships were made of wood, and the ropes woven of papyrus.
You can find more here; the ships were dismantled and stored in the caves. These ships were constructed of red cedar which had been imported from Lebanon.
It is possible to use papyrus boats at sea: Thor Heyerdal managed to do this on his second try, keeping a papyrus raft at sea for two months.
The Afropedia site on Axum ends with some comments on their shipping:
Aksum had a fleet of ships that guarded the Red Sea ports. Aksumite ships were put together by rope fibers. It did not use iron nails to bind wood. Ships sailed as far away as India and possibly China.
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The Kingdom of Aksum is dated from 100β940 CE. The sailing technology of this time frame is way beyond Egyptian papyrus rafts. This begins almost 200 years after the battle of Actium ,one of the great naval battles of the time, and in fact one of the trade partners mentioned is the Roman, and later Byzantine empire.
From wiki page on Indo-Roman trade relations
In the book 'Foundations of an African Civilisation: Aksum and the Northern Horn, 1000 BC ... By David W. Phillipson, on page 200 he discusses that many have assumed the ships were of Roman, or later Byzantine origin. You are looking at a very large time span, after all.
One of the most common Roman merchant ships was the Corbita. These could carry 70-350 tons of cargo! (I cant find a good public domain image, sorry). The ships probably varied along this trade route however, with different type ships operating in the gulf than in the open ocean.
Update: (ignore Roman ships). Finally found some documentation!
"all the boats which are found in India and on this sea (the Red Sea) are not made in the same manner as are other ships. For neither are they smeared with pitch, nor with any other substance, nor indeed are the planks fastened together by iron nails going through and through, but they are bound together by a kind of cording. The reason is not as most persons suppose, that there are certain rocks there which draw the iron to themselves (for witness the fact that when the Roman vessels sail from Aelas into this sea, although they are fitted with much iron, no such thing has ever happened to them), but rather because the Indians and the Aethiopians possess neither iron nor any other thing suitable for such purposes. Furthermore they are not even able to buy any of these things from the Romans since this is explicitly forbidden to all by law" (Procopius, ed. Dewing 1914: 183-4).
This from the book Aksum An African Civilisation of Late Antiquity by Stuart Munro-Hay
This technique of boat construction is called 'sewn boats', and indicates the Aksumites were using boats of Indian design or style for their own vessels.
The best known boats built with sewn construction are the various forms of Dhow native to the Red Sea, Persian Gulf, and Indian Ocean for over two thousand years.