The wounded border guard, Lori Bowcock, was hospitalized in stable condition in Vancouver and was expected to make a full recovery.
The Canadian Broadcasting Corp. reported Crews' identity was confirmed by the British Columbia Coroners Service.
Crews, 32, was trying to enter Canada in a white van with Washington license plates when the incident happened about 2 p.m. Tuesday at the Douglas crossing between Surrey, British Columbia, and Blaine, Wash., the QMI Agency reported.
CKWX-AM, Vancouver, reported Ron Moran, executive vice president of the Canadian Customs and Immigration Union, said two shots were fired.
"There were conflicting versions of what had actually happened at the booth," Moran said. "One of versions was suggesting ... that he committed suicide and it was the same bullet that had basically, accidentally hit the officer. What we know at this stage is that is, was, definitely not the case -- that he opened fire on her. He shot a single shot at her, hit her in the neck. After which he committed suicide."
The southbound border crossing was reopened Wednesday but the northbound crossing was expected to be closed until Thursday, CKWX-AM said.
British Columbia Premier Christy Clark and Washington Gov. Chris Gregoire spoke on the telephone after the shooting and agreed police on both sides of the border will collaborate, the CBC said.
On an average day, about 3,500 vehicles traverse the border at the Douglas crossing, the CBC said.
The CBC said Crews was believed to have been a tattoo artist working in a tattoo parlor in Silverdale, Wash., and that he had moved to Bremerton from Las Vegas.