How can I disable a silverfish spawner?

They actually stop spawning at light level 12 or higher, so surrounding the area with glowstone or redstone lamps should work.

Etho, a relatively famous YouTuber who specializes in exotic mob farms, has been working on a silverfish farm for a while: he went over how to disable a silverfish spawner using only light in episode 155 of Etho Plays Minecraft:

He had the same scenario you have: he wanted to keep the portal accessible but disable the spawner. He goes into possible caveats and problems you might run into, but he was able to completely disable the spawner by adding redstone lamps across the top of the spawning area.

If you want to deactivate a silverfish spawner, it is simplicity once you know it spawns in a 9x9x3 area and requires the light to be above 12. Remember that the portal blocks are modeled after glass and are transparent. This configuration requires 13 JackOLanterns, 3 torches, and some dirt.

Also, it appears that the lava is positioned to where if it were completely released from its nicely contained reservoirs by removing the stairway and pool walls, it should almost light up the room sufficiently, at which point one may want a glass walkway above it.

Happy Crafting

The siverfish spawner mechanics have changed since the accepted answer was written. The current implementation is as follows:

When spawning from spawners, they can only spawn in light level 11 or lower, except on top of stone and its variants, where they can spawn at any light level.

This means that in addition to disabling the spawner with light level 12 or above (as described in the accepted answer), you will have to replace or cover the stone floor and lava pool borders with dirt or wood planks. Don't forget to extinguish the lava if you decide to use wood.