The total number covers the period from January 4th 2016, when the controls were initially put in place, to February 26th this year.
Justice minister Søren Pape Poulsen confirmed the figure in response to a parliamentary question by Danish People's Party immigration spokesperson Martin Henriksen.
Poulsen's ministry obtained the figure from the Danish National Police (Rigspolitiet), Ritzau reports.
Denmark and five other Schengen zone countries are currently preparing a new extension of their border control arrangements by a likely six months.
Current border control measures in Denmark, which is part of the Schengen zone, expire on May 12th.
Immigration minister Inger Støjberg said last week there is no justification not to continue the tightened border control.
"I cannot imagine that we will lift border control at the present time," Støjberg said in Brussels last Thursday, where ministers from the six EEA countries met to discuss extension of the border measures.
When it most recently extended border control in October 2017, Denmark changed its justification for the measure from the high flow of migrants to the country - the original driver for the measure in 2015 - to also citing a stated terror threat.
Støjberg's Ministry of Immigration wrote in a letter to the EU that a “not insignificant” amount of border checks had been carried out, after the union asked for further information at the end of last year.
The border checks take place in two police districts: South Zealand and Lolland, where there are ferry crossings to Germany, and South Jutland, which has a land border.
The two police districts report 941 and 2,964 potentially border control-related charges respectively between January 2016 and February 2018.
Henriksen said that he saw the 5,000 figure as proof that border control is working. The Danish People's Party also wants control introduced on Denmark's border with Sweden.