I closed out a Mutual Fund. My 1099-B contains a cost basis for some shares in the close out and no cost basis for the rest. How does my Turbo Tax Deluxe handle this?

You have two options for how to handle the "no cost basis" entries.  

1. The first option is to enter a zero for your cost basis (because you don't know what it is).  This means you will pay tax on the entire amount you sold those shares for.
2. If you don't like the first option (me either), then you will need to research the value of the shares at the time you purchased them.  If you still have the statement(s) from when you purchased them, you can get the information off those.  Another option is to look them up online.  Yahoo! Finance has a good database of stock and mutual fund quotes.

Once you have the cost basis for all your 1099-B sales, all you have to do is enter them in the Investment Income section of the Wages & Income interview.