Could a space station with thrusters built to it slow down an asteroid to the point where we could control where it goes?
Not unless you have sci fi engines.
Asteroids can be huge. NASA had an asteroid redirect mission that started with the goal of wanting to put an asteroid into orbit around the Moon so humans could visit it. Over time this was scaled back to just picking up a boulder from an asteroid and putting it into lunar orbit since the original goal was impossible. Then the mission was cancelled since at that point it wasn't worth it.
If an asteroid is going to hit Earth and we know about it in advance, we can go to the asteroid and slow it down/speed it up with thrusters. We can't change where the asteroid is going, but we can make it arrive a day later so that Earth is farther along its orbit and is missed.