Yes, the state of New York can suspend an out of state driver's license or privilege to drive in NYS. Whether the state that your license was issued in, your home state, will also suspend your license due to the suspension in NYS will depend upon your licensing state's laws.
In New York you can get your license suspended or revoked for a number of offenses.
Your license will be suspended indefinitely in NYS if you fail to answer a traffic summons or pay a fine (other than parking tickets and fines), surcharge, or crime victim assistance fee and a suspension-termination fee. Your license will also be suspended indefinitely if you fail to file an accident report, submit a bad check for DMV fees, fail to pay support, or fail to satisfy a court judgment that results from a traffic accident. This suspension will be in effect until your correct the condition that led to the suspension.
If you are convicted of operating an uninsured motor vehicle or permitting another person to be operate your uninsured vehicle, your license will be revoked for at least one year in New York. The same penalty applies if the Department of Motor Vehicles receives evidence that you were involved in an accident without being insured.
Alcohol and Drug violation convictions in New York State also usually lead to your license and driving privileges being revoked.
When you are an out of state licensed driver operating a vehicle in a state other than your home, licensing state you are to abide by the driving rules and laws of that state. If you break another state's laws you are normally subject to their penalties just as an in state resident would be.
So if you have an out of state license and have had your license suspended in NYS then you would not be able to drive in NYS until your suspension period is over. Whether you could drive in other states would depend upon what your home state does with regards to the situation in New York. If the offense you were convicted of in NY would get your license suspended in your home state if the infraction of the law had taken place in state, then your state may also suspend your license. This differs from state to state depending upon laws and statutes so you would need to check with your home state's DMV to see if your license is valid or suspended due to the situation in New York.
Keep in mind if your home state suspends your license then it is not valid to use it to drive in any other state. When driving from one state into another they require that you have a valid driver's license, which yours would not be if your home state has suspended it.