The original poster already said that:
1. They don't have a passport.
2. They are unable to get a passport.
There are a number of ways that situation could occur-- for example, if the person is a political refugee who had to flee their country of origin. Not everyone can get a passport by simply applying at their consulate or embassy!
The original poster also said that they are a permanent resident-- not a U.S. citizen. Permanent residents are not eligible to get a U.S. passport.
vt399, this is a question that you really will need to ask your cruise ship company. To avoid the problem of getting different answers from different people, you may want to ask them in writing, so you can get a definitive answer in writing.
Now, there's something called a "reentry permit" or "travel document" (PDF format)
for permanent residents. It's somewhat expensive, $305 plus a $80 biometric fee, and it expires in two years. However, it sounds as if it might be the type of document required in this situation:
You may also want to get a reentry permit if you plan on traveling outside the United States and cannot or do not wish to get a passport from your home country. Many countries throughout the world may allow you to use a reentry permit much like you would use a passport—placing necessary visas and entry and exit stamps in the permit—so you may use it as your main travel document. Be sure to check with any country you plan to visit about specific requirements before you travel.
However, it is unclear whether cruise lines will accept the travel document in lieu of a passport. Again, you'll probably want to check directly with your cruise line.
I'm not a lawyer, and this is not legal advice. However, from what I've been able to determine via searching government sites, it appears that lawful permanent residents of the U.S. can enter the U.S. using only their Permanent Resident Card ("green card"). It also appears that lawful permanent residents of the U.S. can also enter Canada using only their green card. But, I'm not sure if my interpretation of the rules is correct. And, again cruise lines might set their own rules regarding what documents are required.
Edit: grammar: a the -> the