Assuming the original thesis received a DOI to begin with, would it make sense to obtain a new DOI for your own translation of your own thesis, otherwise leaving the content as is, if you published the translation on a preprint server?
A translation of a book gets another ISBN than the original one:
Each different language edition of a book will need its own ISBN.
The same holds true for electronic books.
Regarding your thesis, you should follow this rule for the assignment of DOIs as well. A translation is always a new publication, especially if you publish the translation seperately/delayed on a pre-print server.
Think about it this way: Another translator might also produce a translation of your work. That one would differ from yours because translations depend on the translators. Each translation should have its own DOI to differenciate between them.
You should refer on the title page of the translation or on an extra page the original thesis (DOI) and mention that this ist a translation where you are the translator.
Additionally, Crossref, a DOI registration agency, explicitely states how to relate the DOI of the translation to the DOI of the original work in the metadata deposited along with the DOI: