i don't know of any way to figure out programmatically through DBI whether or not any connections have been made. the only way i can think of is a policy oriented way such as you've suggested (standardizing on $dbh and $sth for handler names).

i checked out the source of DBD::Pg, and from the looks of things, connect() doesn't keep track of connections. perhaps DBD::MySql does, i don't know.