Specifically, is it more efficient to connect once

There is some "setup time" involved with connecting to a database so I'd recommend doing that only once. One of the techniques I use for this involves writing a module that acts as a singleton. Here is a simplified example of that:

package Customer::EmployeeDB;
use DBI;


my $instance = undef;

sub new { 
    if($instance) {
       return $instance;
    } else { 
        bless { dbh=> undef } ,"Customer::EmployeeDB";
        $self->{dbh}=DBI->connect ("DBI:mysql:database=employee","user
+","secret")
                or die $DBI->errstr;
        $instance=$self;
        return $self;
   }

}

sub prepare {
   my ($self,$shift)=@_;

   my $sth=$self->{dbi}->prepare($sql) 
         or die $self->{dbi}->errstr;
   return $sth;

}
1;
[download]

An example of using that module:

  | handwaving here
use Customer::EmployeeDB;

my $db = new Customer::EmployeeDB();

my $sth = $db->prepare("select * from employee order by surname");
$sth->execute();
   | do something with it
[download]

As I said, that's an incredibly simplified version of the actual family of modules I crafted to do the work. I actually go one step further and create modules to encapsulate the queries against the database and those modules instantiate the module encapsulating the database.

Using the singleton behavior ensures that only one connection to the database is made within the entire script. When I profiled the resultant code I was amazed at how much time it took to connect to the database (the real world example was against Sybase, so your mileage may vary) relative to where time was spent elsewhere. The code I had inherited actually made multiple connections to the database and the client wondered why it ran so poorly.