How can I generate a logic pulse from a broken IR beam?
I would start it using a LM567 to detect the presence of someone on the stair. I recommend this circuit: http://www.mondotronics.com/PDFs/3-337_Mod_IR_v22.pdf

Using the output of this circuit, you should trigger your LM555 to keep the light on as much time as you want, but not using a large capacitor as you mentioned, but just a LM555 in monostable operation.

The output from LM555 can drive a transistor to control as many LEDs as you want!

Some level or signal inversion maybe required.