There is a light fixture at the bottom of my basement stairs with a light switch at the top of the stairs. I would like to add a fixture at the top of the stairs to light the stairs. Can I wire right into the switch to add the new fixture?

Maybe and maybe not. It all depends on how the switch is wired. Remove switch plate and switch and tell me what wires you have there.

I have two black wires going into the switch. It is the type of switch where the wires are just pushed into the back of the switch... no screws.

Are there two white wires connected together in the switch box?

Yes, there is a wire nut on those two white wires.

Yes, you may add light. Run 2-wire cable from the switch to where you want the additional light. Connect white wire to the two existing white wires. Connect black to one of the screw terminals on the switch. If you connect to the correct terminal the switch will turn the light on and off. If you connect to the wrong terminal the new light will not turn off. Move wire to the other screw terminal. If light is on 15 amp circuit use 14 gauge wire. If light is on 20 amp circuit use 12 gauge wire

Thank you for the specifics you provided. It is very much appreciated.