How long should a randomly-generated ID string be to remain unique?

I want to generate unique user_id strings...

How long should I make the randomly-generated ID string to make sure that they remain unique enough to avoid collisions as new users are created? It will contain characters a-zA-z0-9.

What's the shortest length I can reasonable get away with for sizes of:

• 1,000 users?
• 1,000,000 users?
• 1,000,000,000 users?

Thanks!