I did this a few years ago, but don't believe the process has managed to get easier at the US Consulate. My following experience / description is for a US father married to a Dominican mother.
The US father has to physically take your child to the Consulate for your appointment in US Citizen Services. (Don't go without the baby in tow, or you will need to return again with the baby). Plan on being there a few hours (in other words, bring a bottle, diapers, wipes and all the goodies to keep Junior happy.) If Mama isn't a US citizen, they won't let her come inside with you.
Bring EVERYTHING. Their forms, filled out. Your US passport and birth certificate. (If you have a new passport since your wife became pregnant, bring along the cancelled passport as well to show the entry and exit stamps during that time). Your legalized wedding certificate. A legalized copy of your spouse's birth certificate. Photos of the two happy parents during courtship, marriage, pregnancy, delivery, new baby, etc. The ultrasound when you found out you were pregnant. Bring your last couple of IRS 1040 tax returns. Bring paystubs if you work in the DR. Bring a letter from your employer verifying your work situation. Bring bank statements. Bring along your last Social Security annual statement showing your contributions into the system over your work history - they are mailed out each year in February. Get photos of your baby at the gate up the sidewalk for the new passport - they will print the correct sizes for the US Consulate to use in the passport.
If you are not married, they will probably require a DNA test at the lab they specify, of you and your child, and then come again on another day with the test results. They might require you to get a DNA test if they have any doubts anyway.
(If you are not familiar with the term "legalized", this means that first, you get an official copy of the Dominican document from the appropriate Dominican Registro Civil - for the marriage certificate this is the office nearest your wedding ceremony, and for the birth certificate,this is the office nearest the place of birth). Sometimes, the hospital will also give you their version of a birth certificate, so bring this also if they gave you one.
To get Dominican documents legalized, you take these documents down to another office behind the Dominican Lottery offices to get a second stamp on the back certifying that yes, this document is real. This process, if you go with all documents needed, will eat up a separate morning getting them legalized.
(ANYONE CAN PIPE IN HERE IF THIS PROCESS HAS CHANGED)
Show up for legalization process at about 6:00 am and you will probably be one of the first 10 in line waiting for the gate to open at 8:00) If you get there early, you will probably finish up by 9:30 or so, too late to then continue on to the US Consulate that day.
For your visit to the US Consulate Citizen Services, you will have a numbered ticket that you first get at the information window inside US Citizen Services, then find a chair to wait.
The waiting area inside will be packed, but some kind soul will probably give you a chair if there are none. If you need to change Junior's diaper, you must exit US Citizen Services, go out to the first waiting area to the bathrooms, change him, and pack up the dirty diaper in your bag) You will be called for your interview when your number comes up.
They will review all of your documents, and then ask their questions. They will review the dates, and cross-reference them against your passport entry and exit stamps to assure themselves that yes, you were here and "had access to the female" in order to become pregnant (my interviewer's words, not mine).
Shorter answers are the best, and no matter what tonter?a they should say, don't let it get to you. Just swallow any bile and shut up. After all, they are the first line of defense in Homeland Security. They will probably ask about your home, job, your partner's history, your partner's work and education history, how you met, details about your wedding, etc.
If everything is in order, and to your interviewer's satisfaction, they will tell you what to do next. If successful, you will return for the baby's passport and the ONE AND ONLY copy of your child's Consular Report of Birth Abroad..
Put this document into a safety deposit box. If you lose it, they WILL NOT replace it. If lost, this means that every time in the future when this is required, you will have to pay about US$60+ to get confirmation of the registered foreign birth.
For some reason, they want to MAIL the baby's S.S. card when it is later issued, but I think if you tell them and repeat it three times, they may hold this for pickup at the Consulate. Otherwise, they have a contracted delivery service counter in the first waiting area, where you will give them your Dominican address and pay them to deliver to your home, and a receipt.
Best of luck.