I re-installed a sink up stairs( it has no vent). I changed nothing but the sink and the p-trap. But when I fill the sink down stairs it backs up into the kitchen sink when I drain it? Talked with the previous owners and they stated that that never happened, but they never filled and drained the sink in the basement before. If I install a vent upstairs will this solve the issue?

But when I fill the sink down stairs it backs up into the kitchen sink when I drain it?
I don't see how that is possible. For that to happen, water would have to run uphill. Assuming of course that the kitchen is above the basement.

Guessing here...

But it's possible that the sink in the basement has a laundry pump on it. If this is the case, then the drain line that the basement sink pumps into (and kitchen sink drains into by gravity) is clogged and the drain will need to be snaked to resolve the issue. Installing a vent will not fix the issue...

Mark

Do you know if your main sewer pipe exits the home through the basement floor or through the wall?