The Lender, in particular, needs to know that you have a steady income and a good history of paying your bills: It gets harder when your HISTORY is not in the U.S. or when you get paid in CASH and do not have a paper-trail for savings and bill-paying.
Good luck and may God bless
If you are looking for a loan / mortgage, you will at least need an ITIN number. They can be acquired for non-residents who are working in the US.
Here is some additional information you might find helpful about banks that will work with non-residents from the Dept. of Immigration and the use of ITIN numbers: