Visual Studio Code has a "Code Fix" that can automatically copies JSDoc into TypeScript annotations. This can be done to an entire file at once.
TypeScript does not come with a minifier. The code it produces is compatible with the target version of JavaScript (e.g. compile ES6 modules into CommonJS) but unminified.
I see. closure-compiler's "advanced" mode takes advantage of jsdoc type info for minifying so I guess generic syntactical minification won't compress as much for the time being.
We (Google TS team) maintain a tool[1] that transforms TS types into jsdoc types for the purpose of feeding them into 'advanced' mode. We also (to answer the grandparent question) maintain a tool[2] that converts Closure-annotated TS into JS. (Why both ways? We transition JS->TS and check it in as the code the user works with, while we use the TS->JS one within the compiler at optimization time.)
I don't know of prod-quality tools to migrate from jsdoc to TS. TS does have comment-level typing, so you can get the benefits without transpiling anything (similar to Flow).
TS pointedly does not minify output. It does the opposite: try to generate code that a human might have written.
It's very easy to incorporate TS into a Babel or Webpack build pipeline though, so you can use a purpose-built minifier of your choice.
The Angular project has a package for translating TS -> Closure, which appears to be used at Google in some capacity. https://github.com/angular/tsickle
It would be really cool to see the user-friendly version (though that might lead to a rabbit hole on the Closure side, too). The Closure compiler remains awesome, just difficult to make use of successfully.
TypeScript does not come with a minifier. The code it produces is compatible with the target version of JavaScript (e.g. compile ES6 modules into CommonJS) but unminified.