Why don't you just use Net::SCP?
(taken from the docs)

  #procedural interface
  use Net::SCP qw(scp iscp);
  scp($source, $destination);
  iscp($source, $destination); #shows command, asks for confirmation, 
+and
                               #allows user to type a password on tty

  #OO interface
  $scp = Net::SCP->new( "hostname", "username" );
  #with named params
  $scp = Net::SCP->new( { "host"=>$hostname, "user"=>$username } );
  $scp->get("filename") or die $scp->{errstr};
  $scp->put("filename") or die $scp->{errstr};
  #tmtowtdi
  $scp = new Net::SCP;
  $scp->scp($source, $destination);

  #Net::FTP-style
  $scp = Net::SCP->new("hostname");
  $scp->login("user");
  $scp->cwd("/dir");
  $scp->size("file");
  $scp->get("file");
  $scp->quit;
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ps. as stated above:
Q: How do you supply a password to connect with ssh within a perl script using the Net::SSH module?
A: You don't. Use RSA or DSA keys. See the ssh-keygen manpage