This is a fairly simple task. I'm working from the Minima template, so your template terminology may be slightly different. You will be looking for the sidebar area in your CSS style sheet.

Look for this in your template.

    .sidebar .widget, .main .widget {
    border-bottom:1px dotted $bordercolor;
    margin:0 0 1.5em;
    padding:0 0 1.5em;
    }

    .main .Blog {
    border-bottom-width: 0;
    }

Then you will replace it with this code:

    .sidebar .widget, .main .widget {
    background:#ffffff;
    margin:1.5em 0 1.5em;
    padding:8px 8px 8px;
    border:1px solid $bordercolor;
    border-bottom:1px solid $bordercolor;
    border-width:1px 1px 1px;
    border-bottom:1px line $bordercolor;
    }

    .main .Blog {
    background:#ffffff;
    margin:1.5em 0 1.5em;
    padding:8px 8px 8px;
    border:1px solid $bordercolor;
    border-bottom:1px solid $bordercolor;
    border-width:1px 1px 1px;
    border-bottom:1px line $bordercolor;
    }

That will give you a nice border around your posts and your sidebar widgets!

You can change your border width editing above code.