Abbott said he would add about 300 Guard members a week to the 250 members whose deployment was announced Friday until the total number reaches at least 1,000 troops.
At a National Guard center in Phoenix, the Governor told the troops they'll make Arizona and the country safer.
Susana Martinez's office said that more than 80 troops would deploy later this week.
Arizona, New Mexico, and Texas all made new pledges that add up to nearly half of the up to 4,000 troops Trump requested.
The one holdout border state is California, led by Democratic Gov.
"So for President Trump to call up the National Guard is not something that's new and unique", Ducey said in defending the decision.
Speaking Monday with reporters after addressing the troops, Ducey said the deployment isn't politically motivated, even as he claimed there is finally an administration in Washington that cares about the border. But Abbott said he wanted to "downplay the notion" that the National Guard would be "trying to take on anybody that's coming across the border".
Despite drops in apprehensions of undocumented immigrants over time, Ducey said March saw a surge of unauthorized border crossings and illegal activity.
"We had a 200-percent increase in traffic across the border", said Ducey.
Although Homeland Security Secretary Kirstjen Nielsen has declined to indicate how long the operation might last, Trump said last week that he expected to keep the units "or a large portion of them, until such time as we get the wall".
He also said he doesn't think the border is a partisan issue. "You show me somebody who is for drug cartels or human trafficking or this ammunition that's coming over a wide-open and unprotected border here".
Trump said last week that he would be sending up to 4,000 troops to the southern border until his proposed border wall was built.
Defence Secretary James Mattis last Friday approved paying for up to 4,000 National Guard personnel from the Pentagon budget through to the end of September.
The troops will not have law enforcement powers but will only provide support. He supports the deployment but is concerned Guardsmen will militarize the border.
El Paso Sector Chief Patrol Agent Aaron Hull says troops are nowhere near deploying yet.
The New Mexico Guard members could help with air support, surveillance and infrastructure repairs, Hull said. "But it certainly is about border security". Evan Westrup, a spokesman for Brown, suggested the Related Press Monday that state officers are nonetheless reviewing Trump's troop request.
Administration officials have said that rising numbers of people being caught at the southern border, while in line with seasonal trends in recent years, require an immediate response.