Some RS232 drivers have a low power shutdown mode which will turn off the port unless it is open/in use. In this case, the TXD pin could float around a little bit, likely around 0V.
So I would say the answer is "it depends on your hardware". You should be able to characterize your PC and see how it works by powering it up and watching it boot.
Laptops are more likely to implement the power save feature. I believe there is a bios or driver option to prevent powering off the serial ports when they are not in use. Not certain when this option kicks in during boot, however. Again, it's a try it and see thing to see if you get a good steady negative voltage from power up.
Good luck,
Mark Stubbs
http://www.bibaja.com