Oct 8: More than 17,000 have arrived in the city from the southern border since April.
Republican states like Texas, Arizona and Florida have been sending migrants to Democratic areas in recent months.
It's part of a row with the White House as unprecedented numbers of people arrive at the US-Mexico border.
Since September, an average of five to six buses have been arriving in the city each day, Mr Adams said at a press conference on Friday. He said that one in five people in the city shelter system is currently an asylum seeker.
Many of those arriving are families with school-aged children and are in serious need of medical care, he said.
The influx is on track to cost New York $1bn (£900m) this fiscal year, and the mayor is calling for federal and state funding to help with the costs.
"New Yorkers are angry," said Mayor Adams. "I am angry too. We have not asked for this. There was never any agreement to take on the job of supporting thousands of asylum seekers."
He added: "The city is going to run out of funding for other priorities. New York City is doing all we can, but we are reaching the outer limit of our ability to help."
The city's social services are "being exploited by others for political gain", he added.
Three states - Texas, Arizona and Florida - have transported migrants to Democratic-led areas, focusing on self-proclaimed "sanctuary" cities that limit their co-operation with federal immigration authorities.
Republican officials in border states say the tactic is aimed at mitigating the impact of migration flows.
They have also said the measure is designed to increase pressure on the administration of President Joe Biden to do more to reduce the number of migrants crossing the southern US border, which has hit a record high this year.
The Democratic-run city of El Paso, Texas, has been offering migrants free rides to New York City and Chicago as a means of alleviating the strain on city resources. El Paso alone has transported more than double the number of migrants - nearly 9,000 - to the two northern cities than have been sent by the Texas governor.
El Paso officials say the rides are voluntary, and that they co-ordinate with the destination cities to help the migrants upon arrival.
As part of his emergency declaration, the New York mayor issued an executive order that allows the city to dedicate resources to support the asylum seekers and expedite any response efforts.
A spokeswoman for Texas Governor Greg Abbott dismissed the declaration by the mayor on Friday.
"The true emergency is on our nation's southern border where small Texas border towns are overrun and overwhelmed by hundreds of migrants every single day as the Biden administration dumps them in their communities," said Renae Eze.
Ms Eze said that, as of October, Texas had sent around 3,100 migrants arriving in the state to New York on more than 60 buses.
