Open Source is a movement of people who want to voluntarily collaborate on and share software. As long as there are people who want to do that, it will continue. I don't see anyone suddenly stopping doing that just because AI exists. Lots of open source projects are banning AI contributions, yet they're chugging along just as before.
Honestly I think we're going to see a lot less AI-written code in the future, and more AI-assistance (PR reviews, documentation, security scans, scaffolding, brainstorming, test suites, etc). Example: to ship one feature for my open source project with a collaborator, we went back and forth for a month to agree on the change, test it, approve it, merge it. The code was pretty tiny. We could get more contributions faster if AI can help us tighten up that lifecycle.