Notice if a student comes late, misses class or is noticeably tired consistently. Many students in financial difficulties work night jobs to pay for school and living costs. However, this can also be an indicator of many other things including have a young child, a substance abuse problem, partying a lot or just being a night-owl.
Take note when a student doesn't have a textbook or has photocopied one from another student. This can indicate hardship as she may not be able to afford her books. If done in a concerned and compassionate way, inquiring about why a student is missing a book can provide an opportunity for the student to discuss her situation with you.
Look for students who wear the same clothes repeatedly. Although this can occur for many reasons, unchanged or frequently worn clothes can be an indicator that a student is skimping on clothing in order to have money for tuition and books.
Watch what students eat. Someone sitting on the campus lawn with instant ramen or macaroni and cheese may be using the strapped college student's standby method for stretching a food budget. Of course, there are people who just like ramen -- so be careful about assumptions without other evidence.
Contact your dean or financial aid office. If you are faculty concerned about a student and feel that information on a student's financial health may be pertinent to your ability to impact them positively, the school may be willing to help you with this information. Since this is not a standard request, you'll need to have convincing arguments as to how your request is in the student's best interest.