不幸的是，:first-of-type只检查标签名（在本例中为tr），并且目前没有:first-of-class伪类。理想情况下，如果有多个.compDetails-row，您可以在运行时修改表标记，如果没有，您最好使用一些JavaScript。
你可以访问CSS/SASS，难道你不能访问JavaScript吗？这个问题只针对CSS，但正如我提到的，如果不修改标记，它实际上是不可行的。
如果您能够访问JavaScript文件，或者在网站的页脚添加JavaScript标记，这里有一个简单的纯JavaScript方法，如果同一个tbody中有多个类，则将multiple添加到.compDetails-row。

// Grab all the `<tbody>` elements in the document as an array
var tbodies = document.querySelectorAll('tbody');

// Loop through the `<tbody>`'s we grabbed
for( i = 0; i < tbodies.length; i++ ){

  // Grab all the `.compDetails-row` elements that exist in the current <tbody>
  rows = tbodies[i].querySelectorAll('.compDetails-row');
  
  // If there's more than one `.compDetails-row`
  if( rows.length > 1 ){
  
    // Loop through the `.compDetails-row` elements
    rows.forEach(function(item){
    
      // Add the `multiple` class to them
      item.classList.add('multiple');
      
    });
  }
}

.compDetails-row {
  position: relative;
}

.compDetails-row.multiple:before {
  content: '';
  display: inline-block;
  height: 100%;
  border-left: 1px solid grey;
  float: right;
  position: absolute;
  right: 20%;
}

<table>
  <thead>
  </thead>
  <tbody>
    <tr class="compDetails-row">
      <td>Row1 Column1</td>
      <td>Row1 Column2</td>
      <td>Row1 Column3</td>
    </tr>
    <tr class="compDetails-row">
      <td>Row2 Column1</td>
      <td>Row2 Column2</td>
      <td>Row2 Column3</td>
    </tr>
    <tr>
      <td>View Details</td>
    </tr>
  </tbody>
  <tbody>
    <tr class="compDetails-row">
      <td>Row1 Column1</td>
      <td>Row1 Column2</td>
      <td>Row1 Column3</td>
    </tr>
    <tr>
      <td>View Details</td>
    </tr>
  </tbody>
</table>

使用the :has() pseudo-class、which is currently a working draft and behind a flag in Firefox，这只能通过CSS实现。

tbody:not(:has(.compDetails-row ~ .compDetails-row)) .compDetails-row::before {
  border: none;
}

JSFiddle

我假设它有一个表作为父级，因此可以使用伪选择器first-of-type，或者如果表没有<thead>，则使用first-child

table tbody:first-of-type .compDetails-row td {
  border: 1px solid red;
}

<table>
  <thead>
  </thead>
  <tbody>
    <tr class="compDetails-row">
      <td>Row1 Column1</td>
      <td>Row1 Column2</td>
      <td>Row1 Column3</td>
    </tr>
    <tr class="compDetails-row">
      <td>Row2 Column1</td>
      <td>Row2 Column2</td>
      <td>Row2 Column3</td>
    </tr>
    <tr>
      <td>View Details</td>
    </tr>
  </tbody>
  <tbody>
    <tr class="compDetails-row">
      <td>Row1 Column1</td>
      <td>Row1 Column2</td>
      <td>Row1 Column3</td>
    </tr>
    <tr>
      <td>View Details</td>
    </tr>
  </tbody>
</table>