package myDBI;

use DBI;

our @ISA = qw(DBI);
sub connect {
 my $proto = shift;
 my $class = ref($proto) || $proto;
 my $dbh = $class->SUPER::connect(@_);
 bless $dbh, "${class}::db";
}

package myDBI::db;

our @ISA = qw(DBI::db);
sub prepare {
 my $dbh = shift;
 my $sth = $dbh->SUPER::prepare(@_);
 $sth->{private_MyDBI_dbh} = $dbh;
 $sth;
}

package main;

my $dbh = myDBI->connect('dbi:Drivername:dbname',
    'user','passwd',{RaiseError=>1});
my $sth = $dbh->prepare("select stuff from table");
my $dbh_copy = $sth->{'private_MyDBI_dbh'};