Can monsters with no speed cross barrier rooms?
Yes.

From the rules, under Special Rooms, page 6:


  Crossing the barrier doesn’t count as moving a space.


and 


  Monsters always ignore barriers. 


The fact that a monster's movement is defined by speed is relevant to discussing how many rooms the monster can move between. 


  At the start of the monster turn, roll a number of dice equal to each monster’s Speed. That’s the number of spaces that monster can move that turn


This means that a monster than doesn't have speed can move 0 spaces each turn. But your question is not related to how many space a monster can move. Your question is about which side of the barrier room the monster is considered to be on. And monsters can choose to be on whichever side of the barrier they want to be on. Going from one side of the barrier to the other isn't "moving" as the game generally describes moving.

It is worth noting that many rules in Betrayal are either unclear or lacking; and this is a case where the rules as written are not fully in line with the thematic answer. That is to say, it makes sense thematically that a monster that "cannot move" couldn't ever get from one side of the tower to the other. Because of this, it makes perfect sense for your playgroup to decide to play the game with the thematic answer rather than the strictly rules-based answer.