Do you mean, “I have a boyfriend and we are dating chastely,” or do you mean, “I have a boyfriend and we are engaging in publicly scandalous mortal sins all the time”?
Generally speaking, a godmother at Baptism or a sponsor at Confirmation should be:
a baptized Catholic who is in communion with the Church,
who is over the age of reason (and usually should have received Confirmation and be over 18),
and who is not engaging in publicly scandalous activity.
So if you were a Mafioso, an abortion provider, or a gay activist who wasn’t promoting same-sex chastity, probably you shouldn’t be a godmother.
If you have a bad conscience over your relations with your boyfriend, then you should talk to him, talk to a priest, amend your life, and go to Confession. You need to set a good example for your godchild.
If you are not sinning, then there is no problem. You do not have to be married to be a godmother or godfather.
You should be willing (and make a plan, and stick to it!) to stay in touch with your godchild no matter where his/her parents move, and to support your godchild’s Catholic faith. You should also pray for your godchild often.
Godparents are important. It is a big spiritual responsibility. But you can grow into it! It is also good practice for having kids of your own.