What are the (dis-)advantages of cooking vegetables in alkaline water?

Is the embolded proposition true?

Yes, the statement is true. Cooking vegetables in an alkaline liquid keeps them bright green.

Why would alkalinity keep vegetables really green?

The reason behind it has something to do with the chemistry of chlorophyl, I don't remember it too exactly, but generally, the chlorophyl molecule doesn't undergo the same changes when heated under alkaline conditions as when heated under neutral or mildly acidic conditions.

Does the embolded proposition suggest cooking vegetables with alkaline water?

I'm not sure what you want to know here, are you asking whether the author used a statement with the intention to induce readers to use the technique? Analysing this would be off-topic.

If so, what pH is best?

Just about any you can create. Slight alkalinity already has an effect (e.g. a pinch of baking soda when blanching), and obviously you can't get into a high pH anyway.

Also, your title asked for disadvantages. The disadvantages are that your vegetables will be softer and mushier, and that they will have an alkaline taste.