Assuming that your PHP code uses only one connection per page (it shouldn't need more), you will never see more MySQL connections than you have Apache MaxClients set.
You should never set Apache's MaxClients higher than your server can sustain. Apache's default settings are designed for a non-PHP configuration, and in many cases or configurations, may be more than your server(s) can comfortably manage.
The trick is to limit MaxClients to what you can manage. If you frequently hit this maximum, consider reducing the keepalive time, or disabling keepalive altogether (not recommended as this increases load on TCP and will make things slower for clients especially if you have lots of images etc).
Memory is usually where the server fails, as if you're using a Unix server with the Apache prefork model (as recommended currently by PHP), a separate copy of PHP effectively loads for each and every client (C code and constant data are still shared, but there is a lot of dynamic data which is created each time).
MySQL does not use a lot of ram for each client, so is not typically a cause of problems.
Of course your mileage may vary on any of this, I recommend developing a proper load testing system on your dev/testing network.
Mark