Lighting, Light Fixtures, Ceiling and Exhaust Fans - Light Fixture old house

How can I install a new light fixture on basement stairs connecting to existing switch? Do I need to connect to switch or to existing light?

Running your new cable to the switch box might or might not work, depending on how it is currently wired. Running your new cable to the existing light, however, will always work.

I have installed the fixture and run wire to existing light, now I need to connect new wires to existing wires. Looks like the existing light fixture is on top of junction box.

My question is should the new light be connected to the old light fixture or the junction box, I am thinking the existing light so it will be controlled by the existing switch. Do I need to cross black to white or anything like that or just connect black to black and white to white.

Your existing fixture should be installed over a junction box. The cable to the new fixture goes into that box. Connect the wires by matching the colors to the existing fixture. Connect the new black by adding it to the connections with the black from the existing fixture. Same for white.