[puppers]

For the host? Definitely! For AirBnB? Definitely, since AirBnb serves not only guests, but hosts as well.

If I can't choose who to allow into my home, I am not using AirBnB as a host. I am confident that almost everyone shares this sentiment.

Imagine if AirBnB required you to let anyone stay at your home. AirBnB wouldn't last another day.

I agree that you shouldn't make judgements based upon only the profile picture. On the other hand, if they post things I find unattractive (like making fun of people of another race), I'm definitely not going to let them in my home. (OP seems to be a victim of the former and I sympathize with him)