border-width in em - but set a minimum border-width

In CSS3, you can try to (ab)use the max css function, if your browser supports it.

border-width: max(1px, 0.1em);
border-style: solid; 
border-color: black;

Unfortunately this awesome CSS3 feature isn't supported by any browsers yet, but I hope this will change soon!

But in CSS2 – no, you can't.
However, you can use JavaScript/jQuery to loop through all elements and increase the border size to 1px.

But this will eat so much performance your browser is gonna crash if you have too many elements on your page (e.g. a table with more than 50-100 rows).
So in other words, no it's not possible.

$("[id$='ReportViewerControl']").find('*')

    .each(function () {

        if($(this).is('#ParametersRowReportViewerControl *')) 
            return;

        //console.log("Processing an element");
        //var cls = $(this).attr("class");

        // Don't add a border to sort-arrow
        if ($(this).is('img')) {
            return;
        }

        var anywidth = $(this).css('width');
        var anywidth = parseFloat(anywidth);
        //console.log("anywidth: " + anywidth);


        //var lol = $(this).css('borderLeftWidth');
        var blw = $(this).css('border-left-width');
        var brw = $(this).css('border-right-width');
        var btw = $(this).css('border-top-width');
        var bbw = $(this).css('border-bottom-width');

        var borls = $(this).css('border-left-style') == "solid";
        var borrs = $(this).css('border-right-style') == "solid";
        var borts = $(this).css('border-top-style') == "solid";
        var borbs = $(this).css('border-bottom-style') == "solid";

        var blw = parseFloat(blw);
        var brw = parseFloat(brw);
        var btw = parseFloat(btw);
        var bbw = parseFloat(bbw);

        //parseInt($(this).css("borderRightWidth"))
        //console.log(parseInt($(this).css("borderLeftWidth")));

        // UpdateLock = true;


        // Set width to 1px where 0px
        if (anywidth == 0)
            $(this).css('width', '1px');

        if (borls && blw == 0.0 || (blw > 0.0 && blw < 1.0)) {
            //console.log("setting border width");
            $(this).css('border-left-width', '1px');
        }

        if (borrs && brw == 0.0 || (brw > 0.0 && brw < 1.0)) {
            $(this).css('border-right-width', '1px');
        }

        if (borts && btw == 0.0 || (btw > 0.0 && btw < 1.0)) {
            $(this).css('border-top-width', '1px');
        }

        if (borbs && bbw == 0.0 || (bbw > 0.0 && bbw < 1.0)) {
            $(this).css('border-bottom-width', '1px');
        }

        // UpdateLock = false;
    });           // End $('*').each

It looks like different browsers handle borders thinner than 1px differently. In Firefox such borders appear to render as 1px wide, but in Chrome they go away. On the other hand it appears that Chrome renders box-shadow even if it's less than 1px, so using it instead of border might be a good idea (Christina was actually suggesting it in the comment to the question). Than again some browsers won't render box-shadow if it's too thin (Firefox does that). Additionally box-shadow does not add the the box model, so using extra margin might be necessary.

Here is an attempt at overcoming those problems. Note that I had to resort to JavaScript userAgent detection hack (just add one extra class to body if it's a webkit browser), because I wasn't able to achieve it using CSS alone.

if (navigator.userAgent.toLowerCase().indexOf("webkit") > -1)
{
    document.body.className += " webkitHack";
}

/* Borders */

.border{
  border:0.0625em solid black;
}

.shadowyBorder{
  box-shadow: 0 0 0 0.0625em black;
  /* Compensating for the fact that box-shadow doesn't count towards box model */
  margin: 0.0625em;
}

.comboBorder
{
  /* If it's not webkit use border */
  border:0.0625em solid black;
}

/* If it's webkit use box-shadow */
.webkitHack .comboBorder
{
  border: none;
  box-shadow: 0 0 0 0.0625em black;
  /* Compensating for the fact that box-shadow doesn't count towards box model */
  margin: 0.0625em;
}

/* Extras */
td, th
{
  padding:5px;
}
.em05
{
  font-size:8px;
}

.em1
{
  font-size:16px;
}

.em2
{
  font-size:32px;
}

.box
{
  display:inline-block;
  height:1em;
  width:1em;
}

<table>
  <tr>
    <th>Font size</th><th>Border</th><th>Box shadow</th><th>Box shadow in webkit</th>
  </tr>
  <tr>
    <td>
      8px
    </td>
    <td class="em05">
      <div class="box border"></div>
    </td>
    <td class="em05">
      <div class="box shadowyBorder"></div>
    </td>
    <td class="em05">
      <div class="box comboBorder"></div>
    </td>
  </tr>
  <tr>
    <td>
      16px
    </td>
    <td class="em1">
      <div class="box border"></div>
    </td>
    <td class="em1">
      <div class="box shadowyBorder"></div>
    </td>
    <td class="em1">
      <div class="box comboBorder"></div>
    </td>
  </tr>
  <tr>
    <td>
      32px
    </td>
    <td class="em2">
      <div class="box border"></div>
    </td>
    <td class="em2">
      <div class="box shadowyBorder"></div>
    </td>
    <td class="em2">
      <div class="box comboBorder"></div>
    </td>
  </tr>
</table>