Personally, I would set RaiseError on the $dbh and put the execute in an eval block. Something like:

my $dbh = DBI->connect(
    $connect_string, $user, $password,
    {
        RaiseError => 1,
        PrintError => 0,
        AutoCommit => 0,
    }
);

sub insertRecord
{
    my ($record, $dbh, $tapeID) = @_;
    
    my %max = getMaxID($dbh);
    
    eval {
        my $sth = $dbh->prepare_cached(addressQuery());
    
        $sth->execute(
            $max{'addresses'},
            ( map { $record->{"subscriber_$_"} qw(
                address_one
                address_two
                city
                state
                zip
                zip4
            )),
            $tapeID,
        );

        ##Execute more queries in the same fashion

        $dbh->commit();
    } if ($@) {
        $dbh->rollback;
        return 0;
    }

    return 1;                   
}
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I also added a few changes to your syntax, which improves the readability. (In my opinion, of course. They are functional-neutral to what you had, which worked just fine.)

Well, mostly worked fine. Your syntax implied that you were passing a reference to the $dbh into your subroutine. There is no reason to do that, as it's already a reference. If you're not doing that, you have a potential error.

Update: Added a sample $dbh definition with the appropriate settings.