To achieve map to bsb conversion, i.e. convert OziExplorer map files (.map) to MapInfo sea chart (.bsb) files, you will have to use some of the dedicated conversion programs, for example the MAPC2MAPC program that seems to support both of these file types just fine.
Convert OziExplorer map file to MapInfo sea chart.
We found one software available in the database for this conversion.
Microsoft Windows software - convert map to bsb on Windows
A digital map conversion software
Convert from map file format
The table with program actions contains information about what each program is capable of doing with their files. This may be a good pointer for further searches for certain file conversions and help you find the converter you are looking for if our search result is insufficient for you. However, it is far from perfect and may sometimes show results which are not really usable because of the nature of the software handles the files. So a conversion that appears as possible, just because of matched actions, may in fact not be possible.