If you are looking for help with
biblatex and Biber for arXiv uploads, have a look at https://github.com/plk/biblatex/wiki/biblatex-and-the-arXiv and Biblatex and arxiv, error. This answer here mainly addresses the question in the title at face value.
In general, the
.bbl file version need not correspond to either the
biblatex or the Biber version. The same goes for the
.bcf version. The
.bcf versions can be stepped up with a
biblatex/Biber release, but they need not be. In practice that means that the versions trail behind the
biblatex/Biber versions and there is only an approximate correspondence between them and the actual package versions.
biblatex 3.5, 3.6 and 3.7 all need
.bbl format version 2.8. The corresponding Biber versions are 2.6 (for
biblatex 3.5 and 3.6) and 2.7 (for
biblatex 3.4 needs
.bbl version 2.7,
biblatex 3.8 has
.bbl version 2.9.
So the newest combination that works with
.bbl 2.8 is
biblatex 3.7/Biber 2.7. As far as I know that is indeed the version arXiv is using after the February 2017 update Biblatex and arxiv, error.
You can find historic releases of
biblatex and Biber at GitHub (https://github.com/plk/biblatex/releases, https://github.com/plk/biber/releases) and SourceForge (https://sourceforge.net/projects/biblatex/files/, https://sourceforge.net/projects/biblatex-biber/files/biblatex-biber/).