Playing slow stairs will help you learn to make your fingers a bit more independent.
In terms of practice, I don't agree with the idea that practising a single pattern over and over is the right way to learn. I think the best way to learn is by playing maps that happen to have stairs of different types in their patterns.
Here are the different kinds of patterns to look for (these names are what I used to refer to these patterns):
ExamplesSimple Stairs
Alternating Stairs
Partial Stairs
Broken Stairs
Trilling Stairs
Each of these patterns helps in a different way. the other key is that you need to practice these patterns at different BPMs, because stairs at low speeds are very different than playing them at high speeds.
Partial stairs are very helpful, because they force you to practice "sections" of stairs one after another, so you get practice starting stairs on each finger.