VBA - put value in each empty row created

Here's a quick and easy and efficient way with only minimal adjustment to your current code.

Sub Insert_Blank_Rows()

Dim rng as Range
Set rng = Selection ' grab top most cell in range, you may want to actually refer to the actual cell.
rng.End(xlDown).Select 'Select last row in worksheet.

Do Until ActiveCell.Row = 1
     'Insert blank row.
    ActiveCell.EntireRow.Insert shift:=xlDown
     'Move up one row.
    ActiveCell.Offset(-1, 0).Select
Loop

'fill blanks with incremental checks
Dim rngBottom as Range
Set rngBottom = Cells(rows.Count,rng.Column).End(xlUp).Offset(1)

Range(rng, rngBottom).SpecialCells(xlCellTypBlanks).FormulaR1C1 = "=""Check""&ROW()/2"

End Sub

I'll throw in this solution, with no looping nor inserting it's very fast (less than 1 second for 20k rows)

Option Explicit

Sub main()
    Dim helperCol As Range

    With ActiveSheet.UsedRange
        Set helperCol = .Columns(.Columns.Count + 1)
    End With

    With Range(ActiveCell, ActiveCell.End(xlDown))
        .Offset(, helperCol.Column - .Column).Formula = "=ROW()"
        With .Offset(.Rows.Count)
            .Formula = "=CONCATENATE(""check"",ROW()-" & .Rows.Count & ")"
            .Value = .Value
            With .Offset(, helperCol.Column - .Column)
                .Formula = "=ROW()-" & .Rows.Count & "+ 0.1"
                .Value = .Value
            End With
        End With
        .Resize(2 * .Rows.Count, helperCol.Column - .Column + 1).Sort Key1:=helperCol.Resize(2 * .Rows.Count), Header:=xlNo
        helperCol.Resize(2 * .Rows.Count).Clear
    End With
End Sub

as per OP's request, it takes move from ActiveCell

So every other row is empty and you want to fill it? One way would be something like

finalRow = cells(1000000,1).end(xlup).row
yourIncrement = 1
for i = 1 to finalRow
    if isempty(cells(i,1)) then
        cells(i,1) = "check" & yourIncrement
        yourIncrement = yourIncrement + 1
    end if
next i

I am assuming your want to fill column 1 (A).

How's this?

Sub Insert_Blank_Rows()
Dim lastRow&, i&

'Assuming column A has the most data (if not change the `1` to whatever column # does have the most data
lastRow = Cells(Rows.Count, 1).End(xlUp).Row

'Select last row in worksheet.
'Selection.End(xlDown).Select  ' Don't use `.Select`
i = 2
Do While i <= lastRow
    Rows(i).Select
    Rows(i).EntireRow.Insert shift:=xlDown
    Cells(i, 1).Value = "Check " & Cells(i - 1, 1).Value
    Cells(i, 1).Value = Cells(i, 1).Value
    lastRow = Cells(Rows.Count, 1).End(xlUp).Row
    i = i + 2
Loop

End Sub

Here, I got one for you. I already tested it and work well for requirement.

Which is special in my code? My code will miss no row. Perfect auto-increment.

And I also reference from BruceWayne's code because I don't want to edit his own code.

Sub checkingData()

    Dim exeRow As Integer 'For indexing the executing row
    Dim lastRow As Integer 'For storing last row

    exeRow = 2 'Checking from first row

    'Assume that First Column has more data row than Other Column
    lastRow = Cells(Rows.Count, 1).End(xlUp).row

    'Loop from First Row to Last Row
    Do While exeRow <= lastRow + 1

        'Select data row
        Rows(exeRow).Select

        'Insert row below data row
        Rows(exeRow).EntireRow.Insert shift:=xlDown

        'Set auto-increment result
        Cells(exeRow, 1) = "Check " & (exeRow / 2)

        'Increase lastRow count because of adding blank row
        lastRow = lastRow + 1

        'Go to next data row
        exeRow = exeRow + 2

    Loop

End Sub