I think the basement idea is just fine. If it is nicely finished and decorated, people will just think of it as "downstairs", not "the basement".
I think the bigger concern might be the pets. Allergies would definitely be a problem for some. Another thing to consider is the behavior of your pets. I know someone with cats and they like to scream sometimes (or whatever the cat term is) when she is behind the massage room door. If you have a dog, barking would be a consideration. Does this dog bark or jump when strangers arrive at the door? Does the dog bark when they see strangers walking by the house? As a client, I would certainly find it disturbing for a dog to begin barking in the middle of my massage.
I had considered a home office for a bit and the set-up is perfect, as clients could come right in the back door and have access to the massage room and the powder room and I have a door that I can shut that prevents access to the rest of the house. I decided against it, as there were too many things I was concerned about:
1. Being sure everything was always clean enough.
2. Having to think about what I was cooking....garlic or onion smells in the home do not make for good "aromatherapy".
3. I was worried about the doorbell ringing in the middle of the massage
4. The sounds of the neighbor's lawnmower in the middle of the massage would have been disturbing to the client
5. What if I forgot to the turn the ringer on the phone off?
6. Liability should someone trip and fall
7. Shoveling the snow in the winter
All of these concerns were in addition to the concerns about strangers coming to the home. What their perception of my professionalism was being as my office was in my home. My own need to get out of the house to go to work somewhere, as opposed to being home all the time.
When I realized there were so many things I was worried about, I decided a home office was not right for me at the time. I am very happy in my office outside of my home. But, now that I am established for 3 years, if I needed to move my office, I would consider moving it to my home. I am established with enough clients who I feel comfortable with and who respect my professionalism that some of the things might not be a worry any longer. Even though, I would still be concerned about many of the things on the list, I have seen that others have made it work just fine and I would have to do things differently at times, but it is definitely workable.
My children are in college, so I did not have to be concerned about what I would do with them if I had clients or their noise level. If you have younger children, this might be something to think about. Additionally, if you are in a neighborhood with young children, the noise level outside and how it travels to your massage space could be a concern.
Good luck!