SKSE requires Steam to be active in the background. Even though you aren't clicking "Play" in Steam, the app must be open. Essential Next Steps
The only safe and official place to download SKSE is the skse.silverlock.org website or the official Nexus Mods page.
Installing SKSE isn't like installing a regular mod through a manager; it requires a manual touch to ensure the "hook" works correctly.
Are you planning to your game from the latest version to 1.6.640, or are you setting up a new mod list from scratch?
By staying on 1.6.640 with the corresponding SKSE link, you ensure: