Halyna Hutchins, 42, was shot while working on the set as director of photography. She was flown to hospital by helicopter but died of her injuries.
The man, 48-year-old director Joel Souza, was taken from the scene at Bonanza Creek Ranch by ambulance.
Police said they were investigating and that no charges had been filed.
A spokesperson for Mr Baldwin, best known for his role as Jack Donaghy on the NBC sitcom 30 Rock and for his portrayal of Donald Trump on sketch show Saturday Night Live, told AP news agency the incident involved the misfiring of a prop gun with blanks.
In a statement to AFP news agency, a Santa Fe sheriff spokesman said Mr Baldwin had spoken to detectives.
"He came in voluntarily and he left the building after he finished his interviews," the spokesman said.
Mr Baldwin is a co-producer of the film and plays its namesake, an outlaw whose 13-year-old grandson is convicted of manslaughter.
The eldest of four brothers, all actors, Mr Baldwin has starred in numerous TV and film roles since the 1980s.
Ms Hutchins was from Ukraine and grew up on a Soviet military base in the Arctic Circle, according to her personal website. She studied journalism in Kyiv, and film in Los Angeles, and was named a "rising star" by the American Cinematographer magazine in 2019.
She was the director of photography for the 2020 action film Archenemy, directed by Adam Egypt Mortimer.
"I'm so sad about losing Halyna. And so infuriated that this could happen on a set," Mr Mortimer said in a tweet.
In a statement, the International Cinematographer's Guild said Ms Hutchins' death was "devastating news" and "a terrible loss".
"The details are unclear at this moment, but we are working to learn more, and we support a full investigation into this tragic event," said guild president John Lindley and executive director Rebecca Rhine.
Police said sheriff's deputies were dispatched to Bonanza Creek Ranch, a popular filming location, at around 13:50 local time (19:50 GMT) after receiving an emergency call about a shooting on the set of Rust.
Incidents such as Thursday's fatal shooting on the Rust film set are extremely rare, but not unheard of.
Real firearms are often used in filming, and are loaded with blanks - cartridges that create a flash and a bang without discharging a projectile.
In 1993, Brandon Lee - the 28-year-old son of the late martial-arts star Bruce Lee - died on set after being accidentally shot with a prop gun while filming a death scene for the film The Crow.
It was later determined that the gun used had earlier fired a round that caused a cartridge to become lodged in the barrel. When blank rounds were later fired, the cartridge was dislodged and released.
Responding to Thursday's news, Brandon Lee's sister Shannon tweeted: "Our hearts go out to the family of Halyna Hutchins and to Joel Souza and all involved in the incident on 'Rust'. No-one should ever be killed by a gun on a film set. Period."