Are making recordings of broadway shows considered illegal?

1) Yes, it is illegal. It violation of the show's intellectual property rights, the rights of the actors and actress to control their own images, rights of publicity, etc.--in short, it violates a number of different rights.

2) You could probably Google  to find some statistics on how many have been prosecuted. The short answer is people have been prosecuted, but mostly if they are trying to resell the tape, display it commercially, or are stupic enough to post it online somehow. It's highly unlikely that if someone did manage to discretely tape it for his own use and only show it at home to his own family that anyone would become aware of the violation--though if they did, they could pursue legal action.