Hi! I have never been thinking about oscillator precision to send / receive data through serial port. Crystal accuracy is so good, that you haven't to take care about your transmission baud rate, I think (if your oscillator is working near crystal frequency).
But maybe I haven't understood your question. You mean about "oscillator frequency changing"? What is your crystal frequency and division factor? What kind of microcontroller are you using?
Regards,
Jacek