Breadboard circuit not working as intended
The inputs of bipolar TTL parts, such as your 74LS08, source current, so need a very low value pull-down resistor to ensure the input is seen as a low.  You don't show the resistors on the schematic, so I'm not sure of their value.

Traditionally, we put the switch between the input pin and ground, with a 5K1 or so pull-up resistor to +5V.  The switch to ground will definitely be seen as a Low.  Unfortunately, this reverses the logic on the switches, so a switch on is a Low, and switch off is High.