If I were a betting man, I would wager that in the next century or two the number of languages in common use will reduce to one or two hundred.
If I were a betting man, I'd take you up on that for several reasons despite your assertion that rapid transit and near instant communication will reduce barriers. (Ignoring, of course, the fact that jet travel is becoming much more expensive and likely less common depending on how how oil supplies do over the next century.)
1. Many languages are spoken in very low-technology places. Most of the languages are also highly local. You mention 6000 languages, but keep in mind their are only about 195 or so countries in the world. That means that some geographic areas have a lot of languages that haven't already been subsumed for one reason or another.
2. We've had jet engines and the internet for decades now, and barriers between people haven't exactly been obliterated. If anything, it's reinforced humanity's natural tendency to seek out others that look/think/act like they do. It's not language related, but let's take a look at U.S. politics: has the internet made conservatives and progressives understand each other's point of view? No, if anything it's encouraged even more separation as it's been easier to go to a website that matches your own personal biases and have them reinforced by others.
3. Language is part of cultural identity for a lot of people. You go tell a Québécois he can't speak French, or tell a Basque he can't speak Euskara. Even in the U.S. the idea of trying to standardize on English as an official language often has racist connotations.
Now, yes, languages are dying out and they're becoming extinct faster and faster. But, to think that we're going to go down to only 200 language in a century? I strongly disagree. If I were forced to guess how many languages would remain, I think a more realistic number, assuming we are starting with 6000, would be about 3000-4000. I think the vast majority of lost languages would be people simply dying out and their descendants learning a more common local language.
Now, I will agree that among the educated and well-connected, we will see a dominant language. Currently, that is English. My guess is that it will remain as such, and become the "lingua franca" of online discussion and international business. But, given that French was previously considered the diplomatic language and is no more, understand that the position of a language is fragile.
Some thoughts from a language geek.