According to the Apple II reference manual (page 100) the Apple II game port only allows a current drain of no more than 100 mA.
Many years ago I made a 6 way switchbox which allowed for 6 joystick/paddles etc. to selected simply by rotating a switch (I'll post DIY instructions on how to make one if there's any interest). To indicate which port was in use I connected an LED (in series with a 100 Ohm current limiting resistor) between pin 1 (+5V) and pin 8 (GND) for each port.
I didn't know what I was doing back then, and the switchbox has worked fine and (as far as I know) hasn't caused any damage to the computer, but ....is having an LED/100 Ohm resistor along with a joystick (or paddles etc.) connected at the same time OK or not? Should I remove the LED for each port or change the resistors to a different value?