You can make good money designing costumes in Hollywood, but the real problem is finding a way to earn a few bucks in between jobs. Most costume designers work on a freelance basis, which means that they are self-employed, and have to seek out their own opportunities from one project to another. There isn't really a Costume Designers 'R Us that a director can visit to go shopping for someone he can hire for his next production.
If you are a freelancer, you will often be paid in three installments—once up front, once when all the work is completed, and the final installment when the film or theatrical production premieres. You may be a residential designer, which is someone who is hired by a theatre company to do pretty much all of their costuming…in that case, you spend so much time in the theatre you're practically a resident. Hence "residential." In this instance you would have a contract with an annual salary. Lastly, you may be an academic designer, who's main job is as a costume design professor at a college. However, you may still act as either a freelancer or quasi-residential designer on the side.
You should also look into USA. Yeah, we know—you already live there. No—USA is also an acronym for United Scenic Artists, the union that represents set and costume designers. They can provide you with pension and welfare benefits, as well as a swell retirement plan, and will ensure that you not get shafted by some shady producer or director who is trying to hire you on the cheap.
It can take a while to reach the top of your craft, so be prepared to put in some very lean years working as a stitcher or pattern maker. If you are designing for theater, you're going to need to find something to supplement your income unless you are one of the few very talented and very in-demand individuals who work on Broadway. If designing for independent film, it's also going to be tough to eke out a living. However, there is a ton of work to be had in Hollywood—even if you can't get a gig as a head costume designer for a little bit, there are many other costumers working in each production's costume department, so if you stick with it and start working for major studios, you can eventually be making 50k or more a year. That's no Wall Street money, but you're drawing and sewing and living on film sets for a living—what do you want?