It is clear that such a thing would be difficult to enforce, and somebody somewhere would still store the blockchain regardless of the potential consequences (a lot of people actually). But it would sure be a pain for anybody with a company to openly use Bitcoin. Think about how pot prohibition works.
Seems to work like this:
- X is banned at some point for arbitrary and nonsensical reasons (extra information in the blockchain or not)
- Availability or price is not affected by the ban.
- Additionally thousand synthetic high powered versions of X are developed and fly under the radar.
Likely, what you have to consider though is in the instance of pot prohibition it doesn't only affect marijuana as a drug but also it's usage as medicine, industrial hemp and using it as a food crop.
A Bitcoin prohibition would have similar deteriorating effects.