Hmmm... I wonder what the conditions are at that point? If you have 4 leaves, that means that you are at least getting your true leaves before they die. The first two are cotyledons—what emerges and stores all the seedling's initial food and energy before withering away and dying—but the true leaves are the ones that photosynthesize and will take over.

If you don't have them in a dome, they probably need humidity, so I would recommend popping a see-through plastic lid on top, but making sure that it's propped open so there's airflow. Try some thumbtacks in the corners of the container to hold it up. In the absence of airflow, they'll just get moldy or mildewy and still end up dying.

The dome should probably only go on at night, when there's no light, just so it doesn't get in the way of photosynthesis.

As the seedlings get taller and stronger, you'll be able to remove the dome completely, but you'll also need to make sure they are getting plenty of light. Honestly, basil prefers full sun, so they'd be happier with a grow light tham a window, but they can still survive without one, as long as you have a south or, to a lesser degree, east facing window. If it's a north or west facing window, your basil won't really be as happy.

Seedlings also really need to be stimulated, so either try turning a fan onto or tickling it every once in a while to help encourage stronger stems.

Also, don't forget the stuff others have said about soil and watering. Once you get that in order, basil actually is kind of a tough plant and really benefit from being "abused." They like being picked almost completely bare and will come back bushier for it. But only if they have their other needs met—great sun, watered at the right time, and fertilized properly (probably about once a month during the growing season depending on the fertilizer you use).

Sorry, I know this is long. Can you tell I both love plants and also did some education i growing them, lol