You need to worry about whether you are the father and an arrest. Too many women are not truthful so one concern would be paternity. By signing the birth certificate you would be acknowledging that the child is yours, subjecting yourself to an admission of your culpability in a statutory rape as well as subjecting yourself to paying child support. Although the vast percentage of statutory rape cases are not prosecuted, depending on how ticked off and forceful the guardians and/or the mother may be in the future, you can still be arrested until the statute of limitation runs.
Mother may get upset that you will not formally acknowledge the child, so it would be in your interest to explain this to her in advance and not wait until delivery day - explain that an acknowledgment would be construed as admission of guilt as to the possible criminal charge. This could but you time - depending on how old she is because more unlikely an arrest will be pursued once the mother turns 18.
You need to speak with two people:
1) A family law attorney, first. There are a lot of issues here — like potential child support, etc. I am not a family law attorney, so go find one.
2) You also might want to speak with a criminal defense lawyer. Much of this depends on the girl's age and whether they do end up deciding to press charges.
This answer is informational only and does not create an attorney client relationship. I am not expressing an opinion on the merits of your case based on the facts above. My answer to this question might seem curt or mean. My job is not to be your friend, but to tell you the truth. Sometimes the truth hurts. You don't want a lawyer that tells you everything you want to hear, pats your head, rubs your belly, and gives you a balloon. That lawyer is not doing you any favors. This is not legal advice. This is free. You pay for legal advice.