I don't know the Minima, nor Processing, but you have three options, really.
- Throttle Processing, or
- monitor Serial.available() and process as fast as you can, or
- throw away commands if you get too many. Basically, dump excessive input from processing, and handle a complete message.
If it was me, first thing I'd do is crank down the baud rate in Processing, because that slows things down "at source". You'd have to be doing a lot, even in an Uno, or do it very poorly, to not be able to handle incoming messages at 9600. If Processing doesn't give you that option, then you're back to fine-tuning your process, AND learning to throw away things in order to keep up. Does processing have any 'software handshake' options, like XON-XOFF, in wherever you setup your serial?