The pros are that she reduces her estate, may remove assets should she need to go on medicaid in the future.
The cons are that your basis in the stock is her cost. You will pay the capital gains tax upon sale. If she held the securities at her death, and transferred them then, your basis would be the value of the stock on her date of death.
If her estate is under $2,000,000 there is no estate tax, the only financial pro is possibly avoiding having a nursing home take the her assets before going on medicaid (the assets must be transferred 60 months prior to medicaid need). An intangible pro is seeing her grandchildren get the to enjoy the assets before her death.
You will paid the capital gains tax if you sell the stock. It will be long term as you also include her holding period.
Your grandmother is not required to file a gift tax return unless the value of the stock exceeds $12,000. Even if the value is higher, there is no tax due unless she has made taxable gifts that total $1,000,000 within her lifetime.