Jail and court officials tell us ... Beasley was booked into Wright County Jail in Minnesota on Tuesday -- ready to serve his punishment stemming from that bizarre gun incident in Sept. 2020.

You'll recall ... prosecutors say Beasley exploded on a family after they had been in his driveway -- alleging the 24-year-old NBA player pointed a gun at the people and ordered them to scram.

Beasley was initially hit with 2 charges over it all -- felony threats of violence and felony drug possession -- but he cut a deal with prosecutors months later to close out the case.

Beasley pled guilty to the threats charge ... and in exchange, the drug charge was dropped and Beasley was ultimately sentenced to 120 days in jail.

Initially, Beasley was set to begin the sentence on May 26 ... though a court official tells us it ended up being pushed to June 1.

Beasley posed for a mug shot when he was booked ... and you can see, he wasn't exactly thrilled to be behind bars.

It's been an eventful last few days for the NBA player ... remember, he just issued a public apology to his wife and family for his fling with Larsa Pippen.