What is the value of a 1935 US 1 dollar silver certificate at the bank?

Banks don't buy or sell collectible currency. Regardless of a bill's age or rarity, a bank is only allowed to give you face value so you would only get $1 for a 1935 silver certificate.

That said, most average-condition 1935 $1 bills aren't worth much more than a dollar or two anyway. There's more information at the question "What is the value of a 1935 US 1 dollar silver certificate?"