First, we don’t actually know if alternate dimensions exist – they’re just part of a theory which helps explain how the Universe formed and what exactly space-time is. But they haven’t been experimentally seen or proven.
I’m also not sure if you’d like to travel to an alternate dimension, if they existed. You’d never know what type of Universe you’d pop out into – maybe in that Universe, everything is made of cheese, or the Earth never formed, or everything is ruled over by giant spider-ants! And you mightn’t be able to come back!!
I’m like Mark: no clue! There’s obviously loads of different takes on it in SciFi, with I think my favourite being in the TV show Fringe (which I highly recommend if you haven’t seen it, great show) where there are weak spots in the material between universes, but in reality, I’m sure if and how it could work…as Mark said, we don’t even know if there are alternate dimensions yet!