"The spiders crawl under the bed."
Translation:A pókok bemásznak az ágy alá.
It changes the meaning, but if you only see the English sentence, you can't tell which one is the current one. The English sentence has two meanings:
The spiders are under the bed, and they are crawling there. This would be translated to "A pókok másznak az ágy alatt", or more commonly "A pókok mászkálnak az ágy alatt".
The spiders are on the floor, and they are going towards the bed with a crawling motion, until they are under it. This is "A pókok bemásznak az ágy alá".
As a native speaker, I think all of these sentences should be accepted.
"Be-" adds a sense of completeness, that is, the action ends at its goal, the spiders will be under the bed. Also, in Hungarian, the space under a bed is something you can get inside or into.
But you could also say "A pókok az ágy alá másznak."
Or: "A pókok másznak az ágy alá." - it has a sense of a current action, in progress.
And the other translation of "under the bed", as PetiM44 points out, is that they are already under the bed, they are crawling around under the bed. In that case, there is no "be-", and it is "az ágy alatt".