Well, they aren't warnings, they are errors. The version of HTML that you are using got rid of many attributes used in HTML tags to format a page. The W3C recommends that almost all formatting of the page, including page layout, be controlled by CSS.
If you would like to keep those attributes in your HTML file, I recommend using a more liberal doctype like HTML 4.01 Transitional. The reason I say this is, like I wrote above, modern browsers force themselves into standards compliance mode, meaning that the bordercolor attribute does not have an affect on most browsers when working in HTML 4.01 Strict mode. Internet Explorer seems to bend the rules sometimes with non-standard attributes when working in Standards compliance mode.
The other reason Netscape borders look bad, and I should have written this earlier, is because the borders are always rendered with the highlight and shadow affect when added through the border attribute in the table tag. This occurs whether or not your force the browser into standards compliant mode.
If the validator lists an errant attribute, simply remove it. You can recreate the affect of the attribute using CSS, and do much more. But if you have several pages to tweak, try using a correct HTML 4.01 Trans. doctype, or use no doctype to force browsers into quirks mode.
Either way, using CSS to set background colors, borders, and font colors and sizes will save you many headaches because it will centralize a good portion of the page layout.
If you want any help with that, we'd all be happy to assist.