Detect RS232 lines activity
You can connect the Rx/Tx lines to something like a 74VHC123AFT (one chip for both lines), and drive the LEDs directly with the output (preferably the /Q output with a series resistor to Vcc). Try a time in the 100ms range. LED current of a couple mA should be sufficient with a good LED.
The retriggerable multivibrator will cause the LED to illuminate continuously if the lines toggle at more than 10Hz. Once the activity disappears, the LED goes off.