You can't make Passive perception checks. Passive perception is what you use when you don't roll any checks; it's the baseline enemies need to beat when trying to hide from you.
In the anti-magic cone situations, the player doesn't check anything. If the beholder is hidden (because its stealth check exceeded the player's passive perception) then the player will not be able to find it.
Whether or not the player detects walking into the anti-magic cone depends on whether something obvious happens when they step into it. For example, if they have a Ioun stone, then they will obviously notice the thing falling to the ground. This does not depend on Passive Perception. On the other hand, if they have no visible magical effects on them, they might not realize that they stepped into an anti-magic field at all.
If the player wants an extra chance of finding where the Beholder is, they can make an active Perception check. That allows them to roll a normal Perception check (not use their passive) but since it requires an Action, they can probably only do that once on their turn. And it still won't help them in finding out about the anti-magic field.