I'm not aware of any security issues. (That does not mean there are none, just that I may be ingnorant. 😉 ) A problem I have seen was a case of a buggy script on another site on a shared host using up all the availalbe MySQL connections via mysql_pconnect, resulting in my scripts failing sometimes due to not being able to connect. Between that experience, the fact that I haven't really seen a need to use it, and the following quotes from the PHP Manual, I have not bothered using it:
[man]mysql_pconnect[/man]:
Warning
Using persistent connections can require a bit of tuning of your Apache and MySQL configurations to ensure that you do not exceed the number of connections allowed by MySQL.
Persistent Database Connections:
Eh...there's too much there I'd want to quote. Just follow the link. 🙂