I like having a CGI::App accessor method, similar to $self->query(), like:

my $dbh = $self->dbh();
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I've also seen a module created just for that, so you say:

use MyDBH;

my $dbh = MyDBH->get();

# or Exporter, but I hate this.
use MyDBH qw($dbh);
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or something. In my opinion the first option is cleaner and more consistent. It seems silly to always say $self->param('dbh'); when you can just define your own dbh() method. More consistent, too (since it is required by your app to function).