UTL_TCP how to monitor number of connections?
UTL_MAIL strongly rely on TCP (Network) and the number of active TCP connections that can be open for a session is 16, per documented Bug 8262564.  Bug 8262564 got fixed in Oracle 11g 11.2.0.1 or later and also on Oracle 11g 11.1.0.7 as one-off patches.  Bug 8262564 is quoted as reference to state that the number of active TCP connections that can be open for a session is 16.  While UTL_MAIL does not have restrictions but the TCP that serves as the base for Mail related packages such as UTL_MAIL and UTL_SMTP does have an active session limit of 16.

You say: "the number of UTL_SMTP connections being made and closed". Are you sure about the closing part? When you open a mail connection, you have to explicitly close it:

--SMTP example
declare
my_conn utl_smtp.connection;

begin
-- open smtp connection
my_conn := utl_smtp.open_connection(
host =&gt; 'www.testlink.com',
port =&gt; 25);

-- Explicitly close the connection oppened
utl_smtp.quit(c =&gt; my_conn);

exception
when others then
raise_application_error (-20002,'An error has occurred.');

end;


Please, note that If UTL_MAIL throws an exception and emails are resend from the exception, the smtp connections are lift open and after 15 successive exceptions UTL_MAIL throws errors.
Connection is getting closed only in those cases when UTL_MAIL.SEND succeeds without any error. Since everything related to a connection is done within the UTL_MAIL package it is not possible to close the connection from outside the package.
I do not know a v$ to count open smpt/tcp connections for an Oracle session.
I think you can do it by yourself, for example using the following code:

set serveroutput on 
alter session set smtp_out_server='&lt;smtp_server_name:25&gt;'; 

declare 
cnt number := 1; 
cnt1 number; 
begin 
for i in 1..20 loop 
begin 
UTL_MAIL.send('valid_email@domain.com','invalid_email@domain.com',NULL,NULL,'testing invalid email','testing invalid email' ); 
Exception 
When Others then 
UTL_MAIL.send('valid_email@domain.com','valid_email@domain.com',NULL,NULL,'send from exp block','send from exp block'); 
dbms_output.put_line(cnt); 
cnt := cnt + 1; 
end; 
end loop; 
End; 
/


I found this code on Oracle Metalink knowledge base. After each error, you increase a counter to monitor the number of open connections.
Regards
Giovanni