Ok, so you drive the 4051 ABC to 000, press button 1L (top left), that puts 5V from 4051-X0 to D13, otherwise R above pulls it low.
Pressing 1R at the same time has no effect as the diode keeps D13 from seeing the the state of 4051-X1 (presumably floating, with not enough current in the high state to drive D13 to a high).
Then you drive ABC to 001, press button 1R, that puts 5V to D13, but since you know ABC=001, it is a different keypress than 1L.
So you are time coordinating the key readings with the state of the 4051 ABC.
I would say 1K is good, you will just need to make sure the power supply can provide (5V/1K = 5mA)* the max number of keys you plan to have pressed at any one time: 10 fingers = 50mA?