Android - How do I disable the USB Connected notification?

If the sound is what you don't like, on some phones the sound can be turned off by disabling USB Debugging. Settings -> Applications -> Development -> USB debugging. However, on some other phones, this will not change the sound that gets played when you plug it in.

To actually remove the notification, as previously mentioned, you'll need a custom ROM to do that for you.

[Update] Another option to disable the sound is to go into your Sounds settings and set the default notification ringtone to silent and then go to each and every one of your applications that you want to have ringtones in and individually set them up there. I don't recommend this option (sure would be a pain if you forgot to set a ringtone to work email and missed an important one saying "We need you here ASAP or you're fired!"), but it is an option nonetheless.

This is part of a Core system in android. It can not be disabled unless you have a ROM that allows you to disable it. I believe 6.x versions of CyanogenMod have/had the option to disable it.

I am not sure if your phone is supported, and I am not sure if that option is still there (since my device doesn't have "official" support currently)