Haven't used Basecamp before, but in my last year of school my program tried using Slack, which seems to be pretty much the same thing; it aims to combine everything we already do - text, e-mails, Dropbox, etc. - in one place.
I found the program easy enough to use, with some fun features, but didn't end up actually using it much. Partly because all our other systems of communication were in place (and working just fine...so no real incentive to change), and partly because we had to use e-mail/Dropbox/whatever as soon as we were working with someone outside our department (which was almost always). In the end, as I recall, we mostly used it as a social/casual group chat, and infrequently at that.
It's designed to be used in an office setting, where you have the same team of people working together all the time, and I can see it being successful in that context. But, as an SM, I didn't find it particularly useful. It just became one more thing I had to check in case someone had left me a message...far less simple/direct than an e-mail or text. (The number of times someone texted me to say they'd sent me a Slack message......arrrgh!)