There are a couple reasons why there doesn't need to be a gate:
- Targaryens had Dragons
- The narrow channel is a death trap.
Targaryens had Dragons
When Dragonstone was founded, the Valyrian Freehold was still thriving, and one of the 40 Dragonlords, the Targaryens, had dragons. When the Targaryens moved to Dragonstone, they brought their dragons with them. There isn't much that can stop the Dragons, especially when an army is strangled on a small staircase. The Dragons would easily burn any invading armies alive, plain and simple.
The narrow channel is a death trap
There is no reason for there to be a gate, a gate there would only be difficult to defend. Any invading army as two options, walk the narrow staircase, or climb the cliffs. Any defences after the staircases are sufficient to get rid of any attackers coming up the staircase. They would have to come up in narrow numbers, 2 by 2, 3 by 3 at most, any archers, dragons, pots of hot oil would easily dispose of any army trying to attack the staircase.
Difficulties in building a gate
So we've covered why a gate is unnecessary, however say they did want to build a gate, the engineering feat would be quite difficult to undertake, and wouldn't be very helpful. The gate would have to be quite narrow, at most two people could stand abreast atop it, and any ships could simply navigate around the gate on the staircase and attack from the side, requiring there to have to be fortifications all around in the ocean to maintain. As opposed to gifting the invaders the staircase and slaughtering them in the narrow space. Furthermore, those couple of people would be isolated from the rest of the defences in the castle (unless you had them pointlessly waiting on the other side of the gate).
The book Dragonstone and the out of universe location
The book Dragonstone looked slightly different:
They may have used the depiction from the books (the only gate being the main gate), combined with the real world location leading them to having a staircase ending with a great castle.