Anyone who's main source of income derives from their playing ,regardless of their actual ability, is a professional bass player.
Someone on here came up with a good definition quite recently along the lines of anyone who earns more money from music than they have spent on their equipment can be described as professional.
I reckon there are three main routes to becoming a professional.
1. Work really hard,become extremely proficient, gain a reputation for flexibility & reliability so that there is a high demand for your skills.
2. Start your own band, work hard, gain an audience get a few lucky breaks.
3. Be in the right place at the right time, join someone else's successful or soon to be successful band.
I reckon all the above apply to the professional function band specialist as much as the all conquering stadium filling rock star.