Will becoming a student staff member affect my financial aid in any way?

Yes.  Being an RA or CEA can affect one’s Financial Aid eligibility in a few ways. First, it can affect one’s overall budget or “cost of attendance.”  To clarify, generally, a student’s total Financial Aid cannot exceed their cost of attendance.  Typically, a cost of attendance includes the following components:
tuition & fees
room & board
books & supplies
some travel
some miscellaneous expenses
A student who is an RA is remunerated with a grant that happens to equal the average cost of University room and board in exchange for their service.  As a result, those components – the average cost of room and board – are removed from their regular financial aid budget.  Thus, if the student were awarded Financial Aid up to the cost of attendance and the Office of Financial Aid is subsequently informed that the student becomes an RA or CEA, the original award would likely have exceeded the new budget and, thus, an adjustment (i.e., partial return of an aid award) would be required.
Take Sharon (a Stern undergraduate) for example: Before she became an RA, her budget or cost of attendance was as follows:
$ 30,430.00 - tuition/fees
$ 16,624.00 - room/board
$ 700.00 - books/supplies
$ 1000.00 - miscellaneous/personal expenses
$ 48,754.00 - budget/cost of attendance
Sharon, if otherwise eligible, can have aid up to $48,754.00, which can include any combination of grants, loans, scholarship and/or work study.  When Sharon becomes an RA, her cost of attendance changes simply by removing the room and board component:
$ 30,430.00 - tuition/fees
$ 700.00 - books/supplies
$ 1,000.00 - miscellaneous/personal expenses
$ 32,130.00 - budget/cost of attendance

Now, Sharon can only have aid up to $32,130.00.  In theory, she wouldn’t need any aid to cover the room and board because she’s receiving a grant that equals room and board fees.  If more aid had been offered and approved before she became an RA, it would now have to be adjusted down (i.e., returned).
Reducing her budget may also affect Sharon’s eligibility for need-based aid.  “Need” is defined by the federal government as her cost of attendance minus her Expected Family Contribution, or "EFC," as determined by Sharon’s Free Application for Federal Student Aid (FAFSA), if she filed one.  Some types of aid – such as Perkins loans, Federal Work-Study, and Federal Stafford Loans – are specifically designed to help a student meet her need.
Keep in mind that one’s EFC always remains the same regardless of the cost of attendance.  Thus, taking the example above, if Sharon had an $8,000.00 EFC, her need would be $40,754.00 ($48,754.00 - 8,000.00 = $40,754.00) if she were not an RA.  She could have need-based aid (if otherwise eligible) for up to $40,754.00.  If Sharon were to become an RA then, again, the room and board component is taken out because it’s already covered by the grant – she obviously shouldn’t need financial aid to help her with that; thus, her cost of attendance is $32,130.00 - $8,000.00 = $24,130.00. Sharon would be eligible only for $24,130.00 in need-based aid.