PuTTY allows you to configure the appearance of the window border to some extent.
The checkbox marked ‘Sunken-edge border’ changes the appearance of the window border to something more like a DOS box: the inside edge of the border is highlighted as if it sank down to meet the surface inside the window. This makes the border a little bit thicker as well. It's hard to describe well. Try it and see if you like it.
You can also configure a completely blank gap between the text in the window and the border, using the ‘Gap between text and window edge’ control. By default this is set at one pixel. You can reduce it to zero, or increase it further.