For June 2019 publishing, stay with 10.2 of course.
11 should be available this fall, with IOS13.
Usuing XCode 11 at this stage is more for preparing (analyse what you want to add to your app, like black mode) and see how to plan modification of code.
So, don't rush today to give up your production code…
As a personal approach, I wait for the second realse (aka 11.1) to move production code.
As you note, testing your app(s) against betas allows you to stretch goal your efforts and be ready when the trigger is pulled on their release, but if you're unfamiliar with such practices, it might be best to take it slow.
Keep in mind you can't submit apps using beta tools/SDKs, so any work against them will be forced to wait for release products.
Any release you have for June would be best kept aligned with current release tools/SDKs, not betas.
Apple will let us know when Xcode 11.x is allowed for the store.
Do use Xcode 11! I found it to be almost unusable and have given up until it is more stable.