What is the cost basis of stocks which are given as gifts?

It depends on the market value at the time of gifting.

If the market value at the time of gifting is higher than the original purchase price then the original purchase price is passed to the recipient.

If the market value is lower than the original purchase price, then the cost basis is a dual basis, and would depend on the price when the stocks are sold in the future. Let me use examples to illustrate:

Let's say A bought a stock at a cost of $10 a share. The market value has dropped to $2. A makes a gift of this stock to B. The cost basis of the stock for B is "dual basis", and it would depend on the price B sold the stock in the future.

If B sells the stock for $15, the cost basis of the stock is $10 (the original cost basis) and B has capital gain of $5.

If B sells the stock for $1, the cost basis of the stock is $2 (the market value at the time the gift was made), so B has capital loss of $1.

If B sells the stock for $7 (or any price between $2 and $10) B has neither a capital gain or capital loss.

The whole idea behind this tax rule is that one cannot transfer capital loss to another person. You will see from the examples above that A's original capital loss of $8 is lost forever.