In dancesport in the US, anyone can be a professional -- even people who have had as little as six weeks of teacher training. All you have to do to be a professional is to say "I'm a professional" and start charging money for lessons. Of course, there are different kinds of professionals.
So the question isn't if you are too old. The question is what kind of professional career do you want? If you want to teach social dancers and wedding couples, you can take a teacher training class with "no experience necessary" and be out there in a few months teaching people how to dance. If you want to be a teacher with higher level students, then you must establish yourself as a competitor so that you can develop the kind of resume that will draw the better students to you.
If you want to be a competitor, ask yourself if you have to be a professional competitor, or if you'd like to try amateur competition first. The barriers for entry into amateur competitions are much lower than for pros. Basically, there are two categories of pro competition, and the people in the lower category are quite accomplished. But for amateurs, there are many more levels and many more opportunities to compete while you learn and improve.
In dancesport, the money to be made is in the high-level teaching, not in the competition prize money and not as a new teacher. (In fact, the real money comes from being a male teacher who takes female students to competitions and competes with them in the Pro/Am events.) Anyway, some studios only pay about minimum wage to new teachers that don't have a lot of experience. And professionals in Pro/Pro competitions might win $1500 or $1000 for first place, but between coaching, costumes, travel expenses, and entry fees, you're lucky if you come out even.
I'm not trying to discourage you, but am rather trying to give a picture of what the professional opportunities out there are. Also, I keep running into people who have no idea that there are amateurs out there dancing in competitions at a high level. In fact, the best amateurs in the US are as good or better than the best pros -- these amateurs are even allowed to earn money teaching and performing. The only real difference at that high a level is what governing body you are registered with.