Electronic – arduino – Where to connect GND on keyboard multiplexer circuit

It depends on final equipment ground policy which may differ. In all systems i designed i use GND star connection, which means the following:

1. All isolated* grounds are connected.
2. There is only one point of connection, near the power supplies.
3. Chassis is connected in the same place with a thick wire.

*there are sometimes grounds that somehow depend on other systems and cannot be connected. Most common: minus of diode bridge in AC rectifier. Another possibility- minus of isolated supply for current sensing, which is connected to a motor phase. But since you are not asking about it, we will not discuss it.

Also on each board near each mounting hole there are capacitors that connect the chassis to grounds in AC domain. This is important, because being thick (in relation to PCB copper planes) the case usually has very low inductance and resistance and may be considered perfect ground for noises of all kinds. But on the other hands you don't want any power to pass through the case, so the connection is by caps.

Speaking of ground policy, nice to see that you use optocouplers. Hope you did not forget power filters.

I'm using a breadboard of questionable quality.

That is what seems to be the source of your problems. Problematic breadboards cause loose connections within the board which you cannot see and hence lead to situations such as open connections etc. I'd try using a different board.

Also the pulldown resistor need not be that high. A 10K resistor works well in most cases. I'd also try switching the IN value from the Arduino instead of connecting it to VCC. I haven't seen the block diagram for the chip but it seems that you may be shorting something internally inside the chip which is why the input drops to 0V after a short time. At the very least, connect a pullup resistor of about 10K between IN and the 5V power source.

Though this may not be very relevant in this case, its important to note that the 4051B is a CMOS device and hence anything less than 1.5V on the input is 0V on the output and the lowermost threshold for a HIGH input is 3.5V. This means that this chip will not work with 3.3V devices without appropriate voltage level translations.