My betta actually dislikes things like hiding caves. She won't go near that portion of the tank. I got 3 different ones to make sure, and kept them there for over a week. She never went to that side of the tank. Lol ol.
Caves or not I think all Bettas will find a safe place they like to rest in. Perseus has live plants, a big silk leaf plant and a decoration with windows in it that he likes to swim though and he can also hide in there which he never does and his safe place is behind or beside the heater on the top of the big silk leaf. Yes I have had caves before also and he would swim in and explore but come right back out after that. They seem safe to me just be extra sure anything with openings are big enough so they cant get stuck.
I don't understand why having water or air trapped in a cave would be dangerous. After all, you want water in there (it's for a fish!), and having an air bubble isn't going to hurt - bettas breathe air anyway. Seems wierd.
Odysseus never really used caves before, but he has a big terracotta pot now that I always find him resting in, when he hasn't burrowed under the leaves I use for substrate. He thinks he's a kuhli loach. :)