My 1099B reports "covered and non-covered" proceeds. Non-covered proceeds do not show cost basis. This results in tax due on non-covered proceeds. Can this be prevented?

Yes.  Brokerage statements will not always report the cost basis and/or sales category, even if the information is available right on the 1099-B. The IRS is receiving the 1099-B information from the broker, but is relying on you to report the correct cost basis and category.

For unreported cost basis:

Make a reasonable estimate, based on old trade confirmations or historical prices from a source such as www.finance.yahoo.com 

For unreported sales category:

If you held the security less than one year, use code B (short-term, noncovered)

If you held the security more than one year, use code E (long-term, noncovered)