<div id="app">
    <div id="app">
      <p> SeletedId's {{ selectedIds }}</p>
      <table class="table table-bordered table-hover table-striped">
        <thead>
          <tr class="text-center">
            <th>
              <span class="badge badge-info">Select</span>
            </th>
            <th>
              <span class="badge badge-success">Name</span>
            </th>
          </tr>
        </thead>
        <tbody v-for="(pay,i) in PaymentType" :key="i">
          <tr>
            <td>
              <input type="checkbox" v-model="selectedIds" :value="{selectedId: pay.Id}" />
            </td>
            <td>{{pay.Name}}</td>
          </tr>
          <tr>
          </tr>
        </tbody>
      </table>
    </div>
  </div>

new Vue({
  el: '#app',
  created(){},
  data: {
    message: 'Hello Vue.js!',
    PaymentType: [{
        Id: 1,
        Name: "Cash",
        HasOtherField: false,
        Remarks: ""
      },
      {
        Id: 2,
        Name: "Check",
        HasOtherField: false,
        Remarks: ""
      },
      {
        Id: 3,
        Name: "Other",
        HasOtherField: true,
        Remarks: ""
      }
    ],
    selectedIds: []
  }
})

<div v-show="selectedIds[option.Id]">
  <span>Remarks</span><input type="text" class="form-control"/>
</div>