How can I see how many MySQL connections are open?

I think there are a couple of ways:

SHOW STATUS WHERE `variable_name` = 'Threads_connected'

or you can do a SHOW PROCESSLIST and find out unique values in the Id column. In old PHP API mysql, there is mysql_list_processes function that does the same as SHOW PROCESSLIST, too.

But first one should work for you. And perhaps you might like to check on other STATUS variables