Typically, yes. As the driver of the vehicle, you are responsible for all occupants of your vehicle.
It depends upon your state laws if you can be cited for a passenger in your vehicle that is over 18 not wearing their seat belt. In some states only the adult passenger would be cited in others the driver and passenger can both be cited for the passenger no using their safety belt.
For instance in New York the driver must make sure that each passenger under age 16 obeys the law. The driver can be fined $25 to $100 and receive three driver license penalty points for each violation. And in the front seat, the driver and each passenger must wear a seat belt, one person per belt. The driver and front-seat passengers aged 16 or older can be fined up to $50 each for failure to buckle up.
In most states passengers over the age of 18 (and sometimes those over the age of 16) are the only ones ticketed if they are a passenger and do not wear their seat belt while in other states law enforcement can cite both the driver and the passenger when the passenger is not using their seat belt.
For example in California Section 27315 of the state vehicle code has three subsections (D, E and F) that apply to citing a vehicle owner, driver and passenger for seat belt violations.
Subsection D states that a person shall not operate a motor vehicle on a highway unless that person and all passengers 16 years of age or over are properly restrained by a safety belt. And then Subsection E goes on to say that a person 16 years of age or over shall not be a passenger in a motor vehicle on a highway unless that person is properly restrained by a safety belt.
So subsection D of this CA Law applies to the driver and basically says he or she can be cited if he or she is not wearing a belt, or is not wearing it properly, or if a passenger is not wearing the belt, regardless of whether the driver is buckled up properly or not. Subsection E applies to passengers 16 or older who are not using the seat belts. We have seen it stated by law enforcement in CA that is it is possible, and quite common, for them to cite both a driver (who was wearing a seat belt) and the passenger when the passenger was not using their seat belt.
To find out your state's laws and if the driver of a car can be ticketed for a passenger not wearing their seat belt contact your Department of Motor Vehicles. They should be able to tell you if you could only be ticketed for the passenger if they are under a certain age (i.e. those under age 18) or for a passenger of any age being unbuckled could result in the driver being cited. If you DMV cannot tell you for certain than try contacting your state's highway patrol who should be able to since they write these kind of tickets on a daily basis.