Mr. Higgins has made plenty of money from his work. He is described as a professor of phonetics, so presumably he holds a university appointment with a salary, but it appears that most of his income is derived from private consulting work—or, more precisely, coaching. He explains this to Colonel Pickering,
This is an age of upstarts. Men begin in Kentish Town with eighty pounds a year, and end in Park Lane with a hundred thousand. They want to drop Kentish Town; but they give themselves away every time they open their mouths. Now I can teach them...
The work from which Higgins makes his money, therefore, is exactly the same as the work he does with Liza, teaching people whose accents betray their origins to talk like the upper classes. It is admittedly difficult to imagine Higgins having the patience and tact for this type of teaching, and we never see him teach anyone but Liza. We also see that Higgins does not appear particularly interested in money. He throws a fairly substantial sum at Liza without a second thought at the end of act 1.
What we see of Mrs. Higgins also makes it clear that Higgins is a gentleman by birth, not just by education and training, so he may well have begun life with a fairly substantial fortune and never have needed to worry about money. This also seems to be the case with Colonel Pickering. Nonetheless, it is quite clear that phonetics is a lucrative "hobby" for Higgins, and he can make plenty of money out of it if he chooses.