Yes, you can seek this through legal process in the court.
Now, the father has a financial duty of support (even if you are not married) and this is a lifetime obligation over the first twenty years or so of the baby's life. So terminating the parental rights can result (but not always) in a loss of this support.
Sometimes where the father has a problem (incarceration, chemical dependency, abuse) some mothers
believe terminating the parental rights is the best option so he won't get near the child. But, a support order
can limit or not allow visitation
with the child until these issues are addressed and the father still has a support obligation. Sometimes abusive partners threaten, "you want money from me, then I get the kid." This is not the law.
Before making this decision you will want to meet with a family law
attorney in your city where the specifics can be reviewed. Look up your local bar association (usually by county) and look for the lawyer referal service. There you can seek out a low cost initial consultation to determine how to proceed. Also consider state support agencies for assistance. These agencies often work with parents to establish a clear support order taking in all of the problems and options. Keep in mind, 18 years is a long time. A non-working father with problems now often evolves into a wage earner, so before abandoning a source of support, look at your options.