Should I ask for generic reference letter? I was given advice to apply for multiple PhD programs because it is difficult to get in.

Some programs require reference letter from professors or supervisors. 
I don’t believe that professors have time and will to give me multiple reference letters for each school I apply to.

I have considered asking them for a generic reference letter.

How would a generic reference letter be received by a graduate school committee?
Faculty members will not write "brand new" letters for every school and program you want to apply for. However, once the basic letter has been written, addressing it to the intended recipient is not very challenging. So it is OK to ask your advisors to write a letter for you, and to send it to multiple places. (If you're applying to schools in the US, the faculty member will be responsible for uploading it herself in any event.)