When I do it I use a 74C914 buffer with a 50K resistor connected to the RS232 signal. These are old parts with if I remember tolerate 25 Volts while operating on a 5 volt supply. They also have nice front end protection. This requires virtually no current from the signal and I can have the LEDs work on either polarity or use 2 and use bipolar LEDs. Try these links for more information: How to get power from RS-232 port and RS232 Voltage Levels - Signals DTR CTS RTS » Electronics Notes.
With RS232 you are using a large voltage which requires current to make the transition plus the capacitance of the wires. This link for the 74C914. https://www.mouser.com/datasheet/2/308/mm74c914-1193419.pdf
Good Luck & Have Fun!
Gil