I have experienced Dropbox connecting slowly, but that is a reflection of the Internet connection after startup. And what you are describing, for me at least, happens after startup; the computer is accessible and usable while Dropbox is doing its thing.
So "startup" might be a bit misleading. The slowdown I experience happens immediately after boot but before the system becomes usable. The Desktop appears, the Dropbox menu item pops up in my menubar, and....my computer is essentially unusable for the next minute or so. In theory, it has finished booting, but in practice I get the spinning beachball if I try to do anything. It isn't until the Dropbox menu icon changes to the green checkmark icon that the computer starts responding.
