Hi Ken,
Seing Fado in Lisbon is a little like seeing flamenco in Spain. Wherever you go, you'll be amongst tourists, but this doesn't mean that it is any less authentic, in the same way as you would go to the Prado in Madrid to enjoy the museum with hordes of other tourists.
That said, there are certainly places where they at least don't rip off the tourist and where the locals go as well. For example, Sr. Fado at Rua dos Remédios 176 is run by a couple called Duarte and Marina. Duarte waits the tables while Marina makes some good homemade Portuguese cooking in the kitchen. Then, after dinner, Marina comes out of the kitchen, drapes her black shawl around her shoulders and starts singing the mournful fado songs while Duarte, having whipped off his waiter's apron, accompanies her on the guitar.
Sr.Fado has shows only a couple of times per week, and has a maximum capacity of 25, so you'll definitely need a booking:
http://www.sr-fado.com/
Just across the street is the Mesa dos Frades. I've never been myself, but I've heard several recommendations of this place:
http://www.mesadefrades.com/
If you walk further into the Alfama Old Town district, located at Rua de São Miguel 20 you'll find the A Baiuca fado restaurant, with tables for around 12-16 diners at the maximum. This is the "fado vadio" variety, which basically means amateur - anybody can take their part in the show and I've been there when a taxi driver has pulled up nearby, got out of his taxi and practically walked straight up to the mic to start singing.
Sr.Vinho in the Lapa district is also highly recommended, although I've never been there myself.
best wishes, Roger.