The answer depends, in part, on the tax status of the 501(c)(3) “foundation” that receives the gift.  As you know, the term “foundation” is not, by itself, legally significant.  The organization could be a public charity, like a community foundation or one of the many “foundations” soliciting gifts to cure a specific disease, or it could be a private foundation, like the Gates Foundation or the Ford Foundation.  The rules are slightly different.  As we have often said, if you don’t know what box you fit in, you don’t know what rules apply.  (See Ready Reference Page:  “What Do We Mean When We Say ‘Nonprofit’?”) 

For financial accounting purposes, it probably does not make a difference.  Ordinarily accountants will account on the financial statements and 990-series tax return for the fair market value of the gift when received on the income statement, the same value that the donor will use when claiming a charitable contribution deduction, and the fair market value at the end of the year on the balance sheet.  If, as you say, the value has appreciated since the donor acquired the stock, the fair market value upon receipt will be more than the donor’s tax basis or cost.

In general, the recipient of a gift takes the cost basis of the donor of the gift, and does not, for tax purposes, get a step-up in basis to the value at the time of the gift.  But a public charity doesn’t have to pay any income tax when it sells the stock, so it doesn’t have to worry about the basis in the hands of the donor.  It pays no tax if it sells at a profit and gets no offset if it sells at a loss.  It doesn’t need to record the basis anywhere.

The rule is different for a private foundation, however.  A private foundation is required to pay a 2% excise tax (occasionally 1%) on its net investment income each year.  Its net investment income includes net capital gain.  Like an individual who receives a gift, the private foundation must measure its gain or loss when it sells on the carry-over basis it received from the donor.  Therefore, in order to pay the proper tax, a private foundation needs to keep a record of the original cost basis of the donor somewhere in order to determine its gain or loss correctly.

To avoid having to pay the excise tax on the gain, the private foundation may want to give the stock to one or more grantees in kind.  Just like the tax treatment of the donor who gave the stock to the foundation originally, a gift in kind is not treated as a sale, so the private foundation would not have to pay the 2% tax on the gain but would still get a full fair market value credit towards its 5% minimum distribution requirement.