I'm not a published cruciverbalist, so I'm not really qualified to answer your question, but in case you don't get an answer here's what I've gleaned from this site and the mailing list:
No, you can't make a career of it.
There are people who have regular constructing gigs like Merl Reagle and Frank Longo and maybe a few others. How do you get those? By constructing puzzles for more years than it takes a doctor or dentist to get through school (and being one of the very best too). There are also long time constructors who gain an editing post somewhere, like Patrick Berry or Ben Tausig. How do you get those? Again, being really good for a really long period of time.
The rest of the cruciverbalists out there, of which there are probably serveral dozen who are regularly published and several hundred who occasionally publish, would fall into the free lance category, making between $50 and $200 a sale for dailies. Even at $200 a sale, you'd have to publish 250 puzzles a year to earn a salary of $50,000 (before taxes!). That's a huge task for a free lancer, given that the number of markets with open submission policies are pretty small. The upshot of that is that the puzzles have to be really good to get published, and if you are just getting started it may be awhile before you can create publishable puzzles.
On the other hand, cruciverbalists and editors seem very helpful and nice and interested in mentoring new constructors (who will become their competition!). So I don't mean my answer to dissuade you from learning the craft - it's a lot of fun and there are a lot of people who are very helpful. But it is a craft, there's a lot to learn, and the market is competitive. So making a career of it seems really more of something you fall into, not something you consciously plan for.