Hope this helps.
First, if the Illustrator image is small, I would enlarge it to fill a letter or even a tabloid page size. Bigger is better in this case because when it's reduced to fit on a screen, it will look even crispier.
Go to File/Document Setup.
In the window that appears,
Click on "Artboard" to show "Printing & Export".
Change the default settings to:
Output - 2540
Gradient Mesh Resolution - 300
Check off both "Split Long Paths" and
"Compatible Gradient and Gradient Mesh Printing"
Now export as jpg with "Maximum" image quality,
"RGB" Color Model and
Resolution Depth of "Custom" "300" dpi
This should look pretty good when previewed on a screen.