Floating input and/or bad (high resistance) jumper. Your hand introduces capacitively coupled mains hum (with your whole body acting as an antennae for it), a non conductive tool does not.
I don't see any decoupling capacitors. There should be a 0.1uF ceramic cap from power to ground directly across each logic chip.
Breadboard jumpers like that are nothing but trouble. They are all far too long for your application, and many suppliers use crappy wire and/or fail to connect the ends properly. Get some scrap CAT5 solid core cable or similar and cut jumpers to the actual length required. Round a single chip all wiring should be nearly flat to the board. Chip to chip can arch up a bit, say 2cm max, and if it is going from one end of the board to the other, its acceptable for it to arch up about 4cm.