Professors at your school should hopefully be sympathetic to the fact that all of your classes are really large. If you ask for a rec, ask a professor you preferably got an A with, or as close to it, I'd call and make an appointment and just say it's regarding the possibility of a recommendation, so they have an idea as to why you're in their office, and bring with you a transcript and a resume, and flesh out the resume with details that would be helpful for them, and aren't usually there. Or else, give them your usual resume, and write a "biography" of your life at college, with achievements, activities, etc, that aren't on your resume. That way, this person that doesn't know you incredibly well will have a wealth of info to work from.
Also, try for professors you've taken for more than one class. And, as an alternative, if you've had grad students teaching smaller discussion sections (which is how it generally works for where I went, BU, another huge school), you can always ask them. Interviews with admissions officials in quite a few books have touched on the fact that a TA's rec is generally as valuable as a professor's, since this person has taught you and worked with you academically in situations just like this, where the prof is teaching 200 students and will never get to know all of them.