In today's environment it will not hurt.
In theory, you are not required to carry a proof of legal residency when travelling to PR or even Hawai. I know because we used to host F1 students in the PRE 9/11 days. The law has not changed but the attitude and flexibility has hardened.
To be safe carry your papers. If you are afraid of losing them because you are going for a wild vacation then as a POOR second choice make color copies of your Passport front page, your US Visa page with the I 94 card and your SEVIS F1 and have them notarized.
Good luck.