I agree here, I open a connection to the database in the constructor and disconnect in the destructor. If your running mod_perl you'll most likely have cached open handles which can be used and you might not need the disconnect, but I do it out of habit. I use code something like this:

use vars qw( %conn );

sub new {
    my $class = shift;
    $self = bless { _dbh => undef, }, $class;
    $conn{'dbname'} ||= DBI->connect( ... );
    $self->dbh( $conn{'dbname'} );
    return $self;
}

sub DESTROY {
    $_[0]->dbh->disconnect() if $_[0]->dbh;
}

sub dbh { @_ > 0 ? $_[0]->{_dbh} = $_[1] : $_[0]->{_dbh} }
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HTH,
Steve