I use a bog standard analogue oscilloscope I was given years ago. It's all you need if you know what you're doing.
The trick with RS232 is to use its protocol to your advantage. There is one special character which, if you send it repeatedly in a tight loop, reveals the baud rate to you. More strictly, it reveals half the baud rate to you.
U
Character U is hexadecimal 0x55, or in binary, 0b01010101. Add that to the start and stop bits, and you get a bit sequence of 1010101010. Repeatedly send that in a tight loop and you get a nice square wave which you can see on the oscilloscope by connecting it directly to either the TX or RX pin depending on if you are transmitting or receiving the U at that point.
Work out the period of that square wave, take the reciprocal of it, and you have the frequency. Double it, and you have the baud rate. Or, if you have a fancy oscilloscope that can detect frequencies, that can tell you precisely what half your baud rate is - just double the figure it gives you.