Do I have to pay taxes on the 1099-B if it was a gift?

The gift is not taxable; however, once gifted any income or gains generated by the assets should be reported by him, and are potentially taxable to him.  Any assets in the account (stocks, bonds, mutual funds, etc.) have the same cost basis that his grandparents did.  If he received a Form 1099-B, and it shows income/gains of $1,100 or more he will have to file and pay taxes. 

When they set the account up when he was 1, they put his name on the account and put a few hundred dollars in every birthday and Christmas but didn't turn the account over to him until he reached college, is this considered as "a gift"?

don't confuse what goes into the account with what comes out of the account.

what goes INTO the account is a gift.

what comes out is potentially taxable.

Let's say grandparents gave $100 each year to the child all the way through high school.  Those are gifts

Let's say the money was invested in Apple stock and by the time the child was ready go to college, it was worth ALOT of money.  The stock was sold to fund the college education.

the broker would issue a 1099-B and the profit on those stock sales is taxable income

make sense? 

Yes, this makes sense, but what is the point of the question in TurboTax? How does it change the taxes owed?

It can potentially change the basis of the stock.  Say the grandparents spent $1,000 on the stock they had originally owned in their name, but then they gave it to your son when it was worth $5,000.  When he sells the stock, he would use the $1,000 as his basis in the stock to calculate his gain.

If they paid $5,000 for the stock, but it was only worth $1,000 when he received it as a gift, then your selling price would determine if you used the amount they paid for the stock or the Fair Market Value at time of gift to determine a gain or loss.