One approach is to create a new stylesheet... or to edit the private stylesheet for the notebook.
For an example of the latter, choose the Format>Edit Stylesheet menu item.
Then in the "Chooose a Style" dropdown menu choose "Output".
Then select the cell bracket of the Output that has been placed in the private stylesheet and choose the Cell>ShowExpression menu item.
In the displayed cell expression change it to
Cell[StyleData["Output"],ShowCellLabel->False]
Then unselect the Cell>ShowExpression menu item and close the private stylesheet.
The result is that your Output cells will no longer have the Out cell lables.
