Would it be a good idea to restrict people from broadcasting transactions if their account reputation goes below a certain value?

Worst idea!
You can't and shouldn't prevent anyone from broadcasting transactions (except if they run out of RC)
Imagine you being downvoted to hell for any reason and find yourself with your funds locked because you can't transact.
Do you really believe the "good folks from the community" (aka curators, meaning they expect curation rewards) will coordinate and waste their precious votes just to unlock you?

The reputation system is flawed and needs to be fixed, replaced, or removed.
Till then, the simplest solution is the front-ends to provide a feature to filter out such low-rep spammy comments.