Can I get married if my boyfriend doesn't have any papers and how can I help my boyfriend to get papers?

Getting a green card for your boyfriend should not be too difficult, if you marry him and file the necessary papers with CIS, so long as he came to the US with a visa, even if the visa has long since expired. But please note that immigration law can be very complicated and applying for legal residence [green card] often involves consideration of several factors which are too detailed to address in this forum. It is in that context that I strongly suggest that you should consult with an experienced immigration attorney.

Once you are married, you can file a petition for him to become a Legal Permanent Resident.

Your fiance needs a current passport (or another current government-issued identification document) in order to get married. In New York, the town or county clerk will not ask about the parties' immigrant status to issue a marriage license. You might, however, want to consult an immigration attorney before getting married because your future husband's ability to become legal in the U.S. depends on many factors, including how did he come to the U.S. (with or without a visa), when did it happen, whether he has any criminal history, whether he attended school in the U.S., etc.

If he entered the U.S. unlawfully, under the current immigration laws, he is not eligible to legalize his status within the U.S. even if he is married to a U.S. citizen. However, he may be eligible for DACA benefits.

You can get married in most jurisdictions as long as you both have valid IDs. To help your spouse get his papers, at a minimum you will need to submit an I-130 petition on his behalf. Depending on the circumstances of his case, he may need to file one or more waivers also.