Strictly speaking there's no rule one way or another, but the most common-sense approach is that if the alternate class still has all the class features that the archetype modifiers/replaces then you can take it.
Ideally you can take any archetype so long as you have what it switches out. This is what Ninja does with Rogue archetypes. So, give up your heavy armor and give yourself the exotic mount at first level in place of a normal mount. Not sure whats PFS legal with it however.
Edit: Oh, just noticed it gives up expert trainer too. Which is a rather lame class feature... Talk with your DM, if its PFS its just out.
By RAW, you would not be able to take the Beast Rider archetype as a Samurai because you would not have the "Expert Trainer" class feature (Samurai instead get "Mounted Archer"); however, if this is for a home game, you might be able to convince your GM to let you take the archetype anyway (replacing Mount and Mounted Archer instead of the standard Mount and Expert Trainer features).
I was under the impression that both Samurai and Ninja were treated as separate classes to their 'parent', and so could not share archetypes?
Edit: Oh, just noticed it gives up expert trainer too. Which is a rather lame class feature...Really? It's the prerequisite for Horse Master, which makes it UBER!
Horse master is definitely good if your multiclassing, but they both get a full mount and boon companion you get earlier and it allows you to do something similar for a few levels. Good for dips!
Don't forget you can almost always talk with your DM.