What are Paige's prospects for her future?

Assuming Paige did not try to go into hiding, her near term situation would have been detention and extensive debriefing. Depending on what she told them and what other evidence they had, it is possible that they would charge her with espionage for her limited involvement in her parents' activities. However, any charges would only be leverage since even her limited knowledge of her parents' tradecraft would be much more valuable than a conviction. The FBI would try to use any carrot and stick available to them to learn about what she knew of her parents, their activities, their associates, locations (like Gabriel's apartment), etc. Knowing her parents were beyond the FBI's reach and given her inexperience, I find it doubtful she would have resisted giving the FBI what it wanted.

Barring possible criminal charges, there really is not much the FBI could do to Paige. She was born in the United States and is therefore a citizen, so they could not legally deport her. Henry would be even safer since there would be no evidence at all of his involvement in his parents activities. He could probably expect an extended grilling, though.

In terms of future prospects, they could kiss goodbye any and all of their parents' assets. Those would be seized and they would basically be on their own. For any kind of easy transition to adulthood, they would need Stan Beeman's help, or that of some other benefactor, though any benefactor could expect to come under FBI scrutiny. The FBI would assume any help from "mysterious" origins was coming from the KGB.

In terms of careers, both of them could forget about ever holding a security clearance (forget military, FBI, state department, or intelligence community). They might also find it difficult to get into law enforcement and even government jobs without security clearances. Politics would be an uphill climb (at least in most districts) since their opponents would have a field day with their background. Other positions of public trust--lawyers for instance--might be hard to get into. This leaves a lot of private sector jobs and of course academia where they might find themselves celebrities.