Minimum ATMega328 speed with UART
The UART clock runs at 16 times the baud rate. So you would could get possibly get away with 19.2 kHz. 1 MHz will be fine.

Most often you would run at a clock rate that gives a convenient divider to generate the UART baud rate clock and is still faster enough for any other processing you need to get done. Each time a byte or packet is transmitted via the UART, put the CPU into a low power mode and wake up again on an interrupt when more data is ready.