It's not clear when they will head to the border.
He said between March 2016 and 2017, arrests at the border were up by 200 percent.
The Texas Tribune reports apprehensions or denial of entry along the border last month increased by about 13,000 people from March of a year ago.
National guard members from Texas were shown in images taken over the weekend at the U.S. -Mexico border.
Brown has repeatedly fought with Trump over immigration protection. Fred Davis, a rancher who lives and works 25 miles from the border, said his community has been "screaming for somebody to do something for 30 years" and is in full support of Trump and Ducey.
Ducey told reporters later that the deployments are needed to stop a surge of border crossings since March and accused the US government of previously ignoring the border "for almost a decade".
Trump said last week he wants to send 2,000 to 4,000 National Guard members to the border. The troops were sent to the town of Marana, northwest of Tucson, according to Ducey's spokesperson, Patrick Ptak.More news: Russian Federation has filed a lawsuit to block Telegram messaging app
The Arizona deployments began Monday.
The Mexican Foreign Ministry has said it was assured by Nielsen that guardsmen would not be armed and would not be making arrests but would be acting in support roles only.
"The idea is there is a mission here and to complete the mission so you can call it successful", Ducey said.
"I don't think this is a partisan issue or an identity issue", he said.
The question of soldiers along the border has raised some concern about militarizing the area.
The deployment will be in order to "support border security measures", a press release said.
"The state of Arizona must not be used as a political pawn that will leave visitors questioning the safety of our state", he said in his letter to Ducey. About 250 troops now are near the border.More news: Apartment fire breaks out in Trump Tower leaves 1 dead
"So it allows us to get more from the Border Patrol to the border", the governor continued.
Ducey indicated that while he'd like to see more "boots on the ground" and high-tech solutions for border security, he's also fully on board with Building The Wall (TM).
Pima County Sheriff Mark Napier said he is comfortable with the Guard's mission.
Ducey spoke Monday as he saw off guard members at the Papago Park Military Reservation in Phoenix. But Napier said any fears have been allayed.
A spokesman for Ducey said Friday that the deployment would be funded under a federal law called Title 32. Federal law restricts the military from carrying out law enforcement duties.
Defense Secretary James Mattis last Friday approved paying for up to 4,000 National Guard personnel from the Pentagon budget through September.
Asked how much the administration expected to allocate to the initiative, Homeland Security said it could not provide an answer, and the Department of Defense did not immediately respond.More news: USA crude oil inventories decrease in past week
"We're dealing with a Border Patrol that's 1,800 agents below its congressionally mandated strength", said Wilson.