As [id://perrin] mentioned, eval is the easiest way to catch these so the user doesn't see a nasty error. However, it might start to seem tedious to put eval blocks around all of your Class::DBI code.

This type of problem is one of the reasons several web app frameworks have popped up on CPAN. CGI::Prototyped, for example, wraps all of the main execution block in an eval. This allows you to build a single error handling sub that knows how to handle the error and display a nice page for the user. This solves the problem not only for Class::DBI 'dies' but allows you to use die throughout your code as a exception.

If you don't want to get into learning a web app framework, you could just steal the idea and put your eval at a high enough level that it catches everything that might die.