Based on your pictures, I would definitely wait until they get their first real leaves. They don't look too crowded from the pics. They can go a bit longer before it being a problem.

Congratulations on all your little babies! :)

Erin

I agree with Erin. What you've got now is the cotyledons. Before you can handle those babies to transplant them they should have some true leaves. Actually, it's sometimes better for the little plants to just thin them. (I know it's hard, and I always feel a bit like a murderer when I do it...) Basil grows so quickly that you might want to consider getting your pots now and just planting your seeds directly into those pots. Put 1 or 2 seeds in each area where you want a plant, and thin the weaker one out. Good luck!