Why does my transistor saturate when I touch the base?
Assuming you're touching nothing else (like, say, the 5V?):
You're a big antenna / capacitive load on power lines / coupling into whatever. It's hard to tell what really is the source of current here.
Together with the base emitter junction, which does act like a diode in this mode of operation, you seem to be  injecting enough current into the transistor to make it operate in saturated region.
That's in fact a bit surprising to me; the BC547C is a relatively low-\$\beta\$ transistor and it takes a bit of current to make the output current large, but here we are. If I had to take a wild guess: your USB supply is not the greatest, it has a lot of ripple at low load, and that's easily coupling into you.
It's harmless to you, but obviously changes the operation of your circuit.
That's exactly why, whenever you have a circuit that has high gain (like your single-transistor amplifier), you need to keep  its input well-protected against such interference: keeping the traces on a PCB short, don't use flying cables, don't use breadboard, always put a low-pass filter before the input to get rid of anything high frequency, maybe shield the box, use differential signaling and make sure any noise coupled in is common mode…