assure that the serial I/O got higher priority
I have seen, I don't know how many, projects where the serial interrupt was set high. The Tx interrupt can, of course be as low as anyone want, the concern is the Rx interrupt. The UART is double buffered, so, if the total processing time of all higher priority interrups that can happen in a 1ms timeframe is less than 1 ms, you can run the UART at 9600 baud at low priority. If tha sum of interrupts in 1ms is more than 1ms, you have real problems. The only case wher you have to up the priority of the serial interrupt is where you have broken the golden rule keep ISRs short and run the UART at a very high speed.

Erik