The Senate voted Thursday by a margin of 59 to 41 to block Trump's declaration that an emergency on the border with Mexico requires the construction of a wall, a project for which Congress has not appropriated money.
Alexander added that he shares the president's goal of border security, but warned allowing Trump to redirect money without authorization could set a unsafe precedent.
The president had threatened a veto of the resolution.
President Trump speaks during a meeting with Ireland Prime Minister Leo Varadkar in the Oval Office of the White House on Thursday.
Trump rejected entreaties from several Senate Republicans to agree to a compromise that would curtail his national emergency powers and instead framed the vote not as a matter of constitutional concerns, but rather as a litmus test on border security.
Trump is expected to issue his second veto in the coming weeks over a congressional resolution seeking to end US backing for the Saudi Arabian-led coalition fighting in Yemen.
Trump wants to use the emergency order to divert billions of federal dollars earmarked for defence spending toward the southern border wall.
Trump thanked Republican senators who voted for his declaration in a Twitter post earlier on Friday. A dozen Republicans defected and joined every Democrat in voting yes.
The declaration of an emergency allowed the administration to access over $6bn in additional funds not appropriated by Congress to build the wall. Last month, roughly 76,000 people tried, including 43,000 children and family members, many fleeing violence and poverty in Central America.
The President was flanked by DHS Secretary Kirstjen Nielsen, Attorney General William Barr and number of other law enforcement officials. The House is not expected to have the two-thirds of the chamber's support needed to override the President's veto.
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