The Tykes were knocked out of the FA Cup in the first round on Saturday by National League side Altrincham.
They have also lost their last seven league games, and 34-year-old Johnson admitted it had been a "shocking run."
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He told BBC Radio Sheffield: "I understand people want me gone. If I do stay - and I want to and believe I will - I know what we require."
Johnson continued: "I have to take full responsibility and I do. It's my place to make changes, whether that be changing personnel or rallying the troops.
"We have to put people out there that we believe will play for the badge on the front and not the name on the back."
The former Oldham boss joined Barnsley in February on a three-and-a-half year contract, but the Reds have struggled to make an impact this season.
They have won just four of their 16 matches, leaving them in 22nd place in League One.
"The whole club has to assess the way to move forward because there's too many losing mentality characters around the place," he added.
"The Under-18s rarely get a win, the Under-21s haven't got enough wins and we haven't got enough wins. Something has to have a clean sweep of the mindset and change."

The shortlist for this year's St David Awards, which will be presented on 10 March, has been announced.
They acknowledge extraordinary achievements, with categories including citizenship, culture and sport.
In the bravery category, Peter Fuller was nominated for stepping in to stop a machete attack in Mold in January 2015.
Matthew James from Pontypridd is also nominated for shielding his partner from bullets during last summer's Tunisia terror attacks.

Sylvia Jenkinson lost one of her closest friends in the explosion at Wood Treatment Ltd plant in Bosley, Cheshire, in 2015.
She said the whole community rallied to help the families of those left behind.
Dorothy Bailey, William Barks, Derek Moore and Jason Shingler all died in the huge explosion.
Her friend Dorothy would have been thrilled by the reaction, she said.
Mrs Bailey, 62, would not have believed the amount of love that flowed towards her after her death, Ms Jenkinson said.
"She would have been really, really happy to realise what people thought about her."
On the day of the fire, Ms Jenkinson had been on her way to a nearby school for a leavers service when she heard a sound like a "big bomb".
She realised the extent of the fire straight away, she said.
She had known most of the people involved in the fire, as had most of the people living nearby, and the fundraising started straight away with everybody joining in.
More than Â£230,000 was raised.
"I feel like I am accepting this award on behalf of Bosley," she said.
She had been absolutely shocked when the envelope with news of the honour arrived, she added.
"I felt thrilled, nervous, humble, privileged - the lot," she said.
"I could not concentrate on anything I was supposed to be doing."
The investigation into the fire continues.

17 May 2017 Last updated at 06:49 BST
Around 9,000 people were killed, and more than 8 million affected by the earthquake - roughly a quarter of the country's population
It also caused massive damage to roads and buildings, with schools closed and people left homeless.
But as time passes things are slowly starting to return to normal, and aid organisations like World Vision are helping people get their lives back on track.
Watch Hayley's report.
Barnsley manager Lee Johnson has asked for time to change a "losing mentality" at the struggling League One club.
[NEXT_CONCEPT]
A man who intervened in a machete attack and a holidaymaker who shielded his fiancee from terrorists have been nominated for awards.
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A fundraiser who helped a community after four people died in a mill fire has been made an MBE in the Queen's Birthday Honours.
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Two years ago, huge earthquakes hit the South Asian country of Nepal.