This claim is currently unproven.
According to a Snopes article written today, which rates this claim as unproven, Chapman cited the CNN article linked below as his source for the security cost and he cited the apparently-correct figure of $148 million as the budget of NEA.
When asked for sources, Chapman cited a New York Times article estimating the NEA's annual funding at around $148 million. (The NEA's website states their 2016 funding as $147,949,000.)
Chapman also cited an estimate first reported by CNN that providing security for the entire Trump family was costing New York City more than $1 million a day
However, the CNN article which he references is citing a figure for protecting all of Trump's family (including his adult children and their families) along with Trump himself while he was the President-Elect and living at Trump Tower, not for the cost of protecting Melania alone (or Melania and Barron alone.) Additionally, later statements from the Mayor of New York City indicate that the cost was actually closer to $500,000 per day, not $1 million per day.
Furthermore, Melania and Barron living in NYC appears to also be a temporary arrangement until the end of Barron's current school year, not a permanent one.
Cost for Security
According to an article published yesterday by New York Daily News:
It cost more than $1 million per day to provide protection for President Trump and his family while he was still living in New York after the election, according to CNN. The city has not announced how expensive protecting just Melania and Barron is.
The cited CNN article is from November 21, 2016, while Trump was the President-Elect and was still himself living in Trump Tower prior to taking office and moving to D.C. It says:
Protecting President-elect Donald Trump and his family is costing New York City more than $1 million a day, according to three city officials.
Additionally, this cost is not just for Trump, Melania, and Barron, but also for Trump's adult children and their families:
Adding to the expense is the cost of police assigned to Trump's adult children and his grandchildren, who are also receiving Secret Service protection, John Miller, NYPD's deputy commissioner of intelligence & counterterrorism, told WCBS Monday. All of them live in the city, and all are entitled to receive Secret Service protection.
On December 7, 2016, Mayor Bill de Blasio of New York City was interviewed by a local TV station. During the interview, de Blasio was asked about these costs:
Lehrer: Well, one of the subsequent rounds will be beyond the inauguration. I’ve read a million dollars a day to protect Melania and Barron Trump, who will keep living here, not in the White House, until the end of the school year. Is that an accurate number? A million dollars a day – who pays for that?
Mayor: I think it’s premature to say that. We put forward a figure for the period from the November 8th election to January 20th. That was approximately $35 million.
November 8 to January 20th is 74 days. $35 million / 74 days is about $473,000 per day. Mr. de Blasio also noted that assuming this would apply to the period after Trump takes office is premature.
Snopes is still waiting on getting more up-to-date information:
We have reached out to the city for more information regarding projected security costs for Melania Trump but have yet to hear back.
Cost for National Endowment for the Arts
According to its own website, the FY2016 budget for the National Endowment for the Arts was just under $148 million.
CNN's original figure of "more than $1 million per day" is indeed more than double $148 million per year. However, at the rate used in the more recent estimate in New York's request for compensation from the federal government, the annual cost of security would be about $173 million per year, which is obviously not double the NEA budget of $148 million. Additionally, New York has not yet indicated what it expects the cost of protecting only Melania and Barron to be vs. the cost of protecting Trump himself in addition to Melania and Barron during the transition period.
Finally, according to an article from The Hill that was published yesterday (Feb 2, 2017):
The White House says first lady Melania Trump will move to the White House at the end of the school year
So, it appears that the situation of Melania living separately from President Trump is a temporary one that is currently projected to end some time around May of 2017. So, even if it still did cost nearly $500,000 per day to protect Melania and Barron, this situation is currently planned to only last several months. As a result, even if it did cost that much per day, it wouldn't cost more than the National Endowment for the Arts on an annual basis due to the situation's limited duration.