If the lines are thin then you need to check for accurate positioning. All artwork designed in Illustrator for video should be created on an artboard with standard square pixel video frame sizes like 1920 X 1080 points or pixels unless you specifically know exactly what you are doing. You need to check the size of the imported artwork in AE and make sure that it is created with even pixel dimensions so that single point lines don't line up half way between the pixel grid. If you plan to scale up the artwork you must not apply raster effects like drop shadows, blurs and other things in AI. This will cause the layer to be imported as pixels instead of vectors. Other than that all AI layers should import properly with Continuously Rasterize working just fine.
A full resolution screenshot with the layer that is giving you problems selected in the timeline and in the Project Panel will tell us a lot more about what may be going on. Select the layer in question in the timeline, press the U key twice to reveal all modified properties, make sure that the Composition Panel is set to a zoom factor of 50% or 100% (200% if you have a HR display like a Mac with Retina display), take a full resolution screenshot and drag it to the Add Reply field on this forum to show us what is going on in your project.