I can only direct you to the paragraph labelled "Thread Safety" in the pod for DBI. From which I'll just quote a selective highlight.

But BEWARE, .... You have been warned.

The passage you quoted is aimed at the authors of modules, not users. Unless you are the author of DBI, and the DB-specific drivers that run on top of it, and you are very knowledgable about the internals of the vendor-supplied DB interface drivers, you should probably not be trying to "make my DBI thread safe". Accept the limitation and consider other ways of tackling your problem.

The first thing to consider is: why do you want to access the database from multiple threads in the first place? If you can describe the application that you envisage using this for, then it may be demonstrable that there would be no benefit to having multiple threads accessing the DB. It may be better to have a single thread that fetches the data and then mutiple threads to process it.

Without specific scenarios, it's not at all clear whether there would be any benefit to doing this.