The default behavior of responsive datatable with checkbox srow selection use the same columsn for expansion and select.

To resolve, you assign a separate column (first column) to handle the responsiveness, and a different column (2nd column) for row selection.

Here’s an example. https://www.titanwolf.org/Network/q/f8b83600-61df-49b0-92de-15625240b4b0/yQuestion Responsive datatable with first column as checkboxwww.titanwolf.org

Some extra explanation on how to have the responsive (+ sign) handling in a separate column. https://datatables.net/extensions/responsive/examples/child-rows/column-control.html

$(document).ready(function() {
  var table = $('#example').DataTable({
    select: {
      style: 'multi',
      selector: '.select-checkbox',
      items: 'row',
    },
    responsive: {
      details: {
        type: 'column',
        target: 0
      }
    },
    columnDefs: [{
        targets: 0,
        className: 'control'
      },
      {
        targets: 1,
        className: 'select-checkbox'
      },
      {
        targets: [0, 1],
        orderable: false
      }
    ],
    order: [2, 'asc']
  });
});

<script src="https://code.jquery.com/jquery-1.12.0.min.js"></script>
<link rel="stylesheet" type="text/css" href="https://cdn.datatables.net/1.10.11/css/jquery.dataTables.min.css" />
<link rel="stylesheet" type="text/css" href="https://cdn.datatables.net/buttons/1.1.2/css/buttons.dataTables.min.css" />
<link rel="stylesheet" type="text/css" href="https://cdn.datatables.net/responsive/2.0.2/css/responsive.dataTables.min.css" />
<link rel="stylesheet" type="text/css" href="https://cdn.datatables.net/select/1.1.2/css/select.dataTables.min.css" />

<script type="text/javascript" src="https://code.jquery.com/jquery-2.2.0.min.js"></script>
<script type="text/javascript" src="https://cdn.datatables.net/1.10.11/js/jquery.dataTables.min.js"></script>
<script type="text/javascript" src="https://cdn.datatables.net/buttons/1.1.2/js/dataTables.buttons.min.js"></script>
<script type="text/javascript" src="https://cdn.datatables.net/buttons/1.1.2/js/buttons.flash.min.js"></script>
<script type="text/javascript" src="https://cdn.datatables.net/buttons/1.1.2/js/buttons.html5.min.js"></script>
<script type="text/javascript" src="https://cdn.datatables.net/responsive/2.0.2/js/dataTables.responsive.min.js"></script>
<script type="text/javascript" src="https://cdn.datatables.net/select/1.1.2/js/dataTables.select.min.js"></script>

<div class="container">
  <table id="example" class="display nowrap" width="100%">
    <thead>
      <tr>
        <th></th>
        <th></th>
        <th>Name</th>
        <th>Position</th>
        <th>Office</th>
        <th>Age</th>
        <th>Start date</th>
        <th>Salary</th>
      </tr>
    </thead>

    <tfoot>
      <tr>
        <th></th>
        <th></th>
        <th>Name</th>
        <th>Position</th>
        <th>Office</th>
        <th>Age</th>
        <th>Start date</th>
        <th>Salary</th>
      </tr>
    </tfoot>

    <tbody>
      <tr>
        <td></td>
        <td></td>
        <td>Tiger Nixon</td>
        <td>System Architect</td>
        <td>Edinburgh</td>
        <td>61</td>
        <td>2011/04/25</td>
        <td>$3,120</td>
      </tr>
      <tr>
        <td></td>
        <td></td>
        <td>Garrett Winters</td>
        <td>Director</td>
        <td>Edinburgh</td>
        <td>63</td>
        <td>2011/07/25</td>
        <td>$5,300</td>
      </tr>
      <tr>
        <td></td>
        <td></td>
        <td>Ashton Cox</td>
        <td>Technical Author</td>
        <td>San Francisco</td>
        <td>66</td>
        <td>2009/01/12</td>
        <td>$4,800</td>
      </tr>
    </tbody>