If you live in Europe or Asia, sponsors do exist. Most of the high-level pros and amateurs are sponsored to some degree or another. Not all are fully sponsored to the point of not requiring any other type of income, but some are.
Of course, finding these sponsors is another story. Most of the time, they find you... when you've proven yourself as a good representative of their company. In other words, you have to already be good. Otherwise, you're left to your own devices to find sponsors, which can be very difficult.
If you live in the U.S., finding sponsors is even more difficult. And of the existing sponsors, very few (if any) actually provide enough money to make a living from.
Most U.S. professional competitors use the "Pro-Am" system to support themselves through competitions. They use money they make from teaching to finance their competitive urges. They also take students to competitions, because it it customary for the student to pay the teacher's expenses. If you have a really good student, it alleviates the cost of hotels, airfare, etc. Of course, you still have to pay for your own entry fees and coaching lessons, but it still helps a lot.
To be honest, I have no idea how amateurs in the U.S. get along. I think they just pay their own way to everything. I imagine it's quite difficult.
Sincerely,
Jonathan Atkinson www.ballroomdancers.com