You will almost always want to use shared libraries over static libraries. A key advantage to using shared libraries is that if the library is updated, you can replace the shared libraries with the newer version (assuming binary compatibility) and reap the benefits of the improved implementation without recompiling your application. Additionally, using shared libraries saves space, if multiple programs are using them.
As for the dependencies issue, it is possible to link against a specific version of a shared library, or to place your shared libraries in a special location that is specific to your program -- which doesn't save you space, but which does give you the flexibility associated with shared libraries -- so that should not be a reason to choose static libraries over shared libraries. I am actually hard pressed to come up with a single instance, on a typical desktop, laptop, or server machine where using static libraries is better than using shared libraries.
P.S. If you are trying to install Boost on Ubuntu Linux, just run "sudo apt-get install libboost1.37-dev". You were probably getting errors because you did not install all of Boost's dependencies. These are automatically downloaded and installed when you use Ubuntu's apt-get package manager to install it. Also, it is generally better to use an OSs package manager for installing software packages, than to build from source. For example, using the package system's version of Boost will make it more likely that your software will run smoothly on other Ubuntu Linux deployments which use the package manager's version of Boost.
P.P.S. Boost uses some very advanced features of C++. It kind of pushes C++ to the limit. It is not uncommon to see warnings when compiling Boost. In fact, I have built Boost quite a number of times on various operating systems, and I don't recall a time when there were no warnings.