Gardner McIntosh was driving a Lothian region transport bus on the number 4 route in Edinburgh when the boy boarded the vehicle on February 13 in 2017 at a stop at Oxgangs.
Mr Gardiner said he noticed the boy with a large rucksack as he pulled up at a bus stop in the city.
The bus driver said: "When he did get on the bus and presented his ticket he was unsteady on his feet at the time. Obviously I realised there was some issue at that point." He said the boy said he was "a bit drunk".
"Subsequently I learnt he was 13. I would have put him at late primary school age," he told the High Court in Edinburgh.
Mr McIntosh said he contacted his control room and explained he had a boy he thought was 11 or 12 on board who seemed to be under the influence and required assistance to get to where he wanted to go.
He said: "I said I felt he was too young and he was not in a good state."
Mr McIntosh said: "You always have particular concerns over youngsters, children or elderly people, perhaps with dementia,, things like that."
He was expecting a van to meet the bus but in the meantime another member of the bus firm joined the bus and he asked him to go to speak to the boy.
During the journey he received a call on the radio during which a boy was named. He said: "Information was there now he had been missing for days and the police had been looking for him everywhere."
The boy had got off the bus with the other staff member but started to slip down at the side of a bus shelter.
Mr McIntosh, 51, was giving evidence at the trial of Derek McNeill, 52, who is accused wilfully neglecting, exposing and abandoning missing person and diabetic Blake Ross, 13, in a way likely to cause him unnecessary suffering or injury to health between February 11 and 13 in 2017.
He is alleged to have failed to seek or attempt to seek medical attention and treatment despite the boy's deteriorating medical condition and abandoned him unsupervised on a public service vehicle in Edinburgh.
It is said that Blake did not have access to timely medical treatment and developed ketoacidosis and died at the Royal Hospital for Sick Children in Edinburgh from complications on February 13 in 2017.
Christopher Borowski, a residential care officer, who acted as Blake's key worker at the unit, previously said the teenager was in receipt of two types of medication. One was taken once a day but the other he would take about five times daily before meals.
McNeill has denied the offence under the Children and Young Persons (Scotland) Act. He has also denied committing a sexual offence and attempting to pervert the course of justice.
The trial continues.