State laws differ so in some states it may go on your record while in others it will not since you were a passenger. The fine amount will again differ by state and even possibly be determined by the jurisdiction in which you received your citation.
You did not mention if you were the driver and received this ticket due to a passenger not wearing the ticket or were instead the passenger that was not in the seat belt. If you were the driver then points may be placed against your driving record in many states while if you were the passenger it is less likely that points will be assessed or that it would be placed on your driving record, however again state laws differ.
In California you may not operate your vehicle unless you and all your passengers 16 years of age or over, or who weigh 60 lbs. or more, are wearing seat belts. If seat belts are not worn, you may be given a traffic ticket. Also, the driver will be given the ticket if a passenger, younger than 16, is not wearing his/her seat belt.
This offense in CA is punishable by a fine of not more than twenty dollars ($20) for a first offense, and a fine of not more than fifty dollars ($50) for each subsequent offense. We have read though that with courts costs, etc a first offense fine usually ends up being around $85.
In Texas drivers can be stopped and ticketed for a safety belt violation if they, or any of their passengers under the age of 17, are not buckled up or properly restrained. Traffic fines for failing to use a seat belt or secure a child in a safety seat can run from $50 to $200, plus court costs.
It appears that you may be living in New York. According to the New York State Department of Motor Vehicles, NYS law requires all front seat passengers to wear seat belts. Children under the age of 16 must wear seat belts when they are in the front seat or the back seat. Children under the age of four must ride in safety seats. Children age four, five or six must ride in child restraint systems.
The NYS DMV states that the penalty for a seat belt or car seat violation is a fine of up $50 (plus court costs, administrative fees, etc). If a conviction includes a person under the age of 16, the driver receives a maximum fine of $100 and 3 points on the driver license record.
If a driver is convicted of a violation where a passenger under age 16 is not wearing a seat belt or in a child safety seat, 3 points are assigned to the driver's record. If convicted of a seat belt violation involving the driver or a passenger age 16 or older, no points are assigned to the driver's record. So there is the points situation if you were the driver and received a NYS traffic ticket for allowing a passenger to not have their seat belt buckled up.
If you have received a ticket as a driver or as a passenger that was not wearing the seat belt / safety belt call the court listed on the ticket to find out your exact fine amount. The court or your state's Department of Motor Vehicles should be able to tell you if this violation will go on your driving record and if points will be assessed.