I am using Python3.6 and MacOS 10.13.2(Beta) but should work with similar setups.

First create and activate your virtual environment, by default mine is created on top of Python3.6.

~ ❯❯❯ mkvirtualenv basemap-virtualenv
[...]
Installing setuptools, pip, wheel...done.
[...]
(basemap-virtualenv) ~ ❯❯❯ cd Desktop
(basemap-virtualenv) ~/Desktop ❯❯❯ curl https://codeload.github.com/matplotlib/basemap/tar.gz/v1.1.0 -o basemap-v1.1.0.tar.gz

Note that I am using the latest version 1.1.0 as of the time of writing this, but you should use the latest available on https://github.com/matplotlib/basemap/releases.

Let’s start the installation

(basemap-virtualenv) ~/Desktop ❯❯❯ tar -xvf basemap-v1.1.0.tar.gz
(basemap-virtualenv) ~/Desktop ❯❯❯ cd basemap-1.1.0/geos-3.3.3/
(basemap-virtualenv) ~/D/b/geos-3.3.3 ❯❯❯ export GEOS_DIR=/usr/local
(basemap-virtualenv) ~/D/b/geos-3.3.3 ❯❯❯ ./configure --prefix=$GEOS_DIR
(basemap-virtualenv) ~/D/b/geos-3.3.3 ❯❯❯ make; make install

After this geos library is installed, let’s jump onto installing basemap.

(basemap-virtualenv) ~/D/basemap-1.1.0 ❯❯❯ pip install numpy
(basemap-virtualenv) ~/D/basemap-1.1.0 ❯❯❯ pip install pyproj
(basemap-virtualenv) ~/D/basemap-1.1.0 ❯❯❯ pip install pyshp

(basemap-virtualenv) ~/D/basemap-1.1.0 ❯❯❯ python setup.py install
[...]
customize UnixCCompiler
customize UnixCCompiler using build_ext
building '_geoslib' extension
compiling C sources
C compiler: clang -Wno-unused-result -Wsign-compare -Wunreachable-code -fno-common -dynamic -DNDEBUG -g -fwrapv -O3 -Wall -Wstrict-prototypes

compile options: '-I/usr/local/include -I['/Users/myself/.virtualenvs/basemap-virtualenv/lib/python3.6/site-packages/numpy/core/include'] -I/Users/myself/.virtualenvs/basemap-virtualenv/lib/python3.6/site-packages/numpy/core/include -I/usr/local/Cellar/python3/3.6.2/Frameworks/Python.framework/Versions/3.6/include/python3.6m -c'
clang: src/_geoslib.c
zsh:1: no matches found: -I[/Users/myself/.virtualenvs/basemap-virtualenv/lib/python3.6/site-packages/numpy/core/include]
zsh:1: no matches found: -I[/Users/myself/.virtualenvs/basemap-virtualenv/lib/python3.6/site-packages/numpy/core/include]
error: Command "clang -Wno-unused-result -Wsign-compare -Wunreachable-code -fno-common -dynamic -DNDEBUG -g -fwrapv -O3 -Wall -Wstrict-prototypes -I/usr/local/include -I['/Users/myself/.virtualenvs/basemap-virtualenv/lib/python3.6/site-packages/numpy/core/include'] -I/Users/myself/.virtualenvs/basemap-virtualenv/lib/python3.6/site-packages/numpy/core/include -I/usr/local/Cellar/python3/3.6.2/Frameworks/Python.framework/Versions/3.6/include/python3.6m -c src/_geoslib.c -o build/temp.macosx-10.12-x86_64-3.6/src/_geoslib.o -MMD -MF build/temp.macosx-10.12-x86_64-3.6/src/_geoslib.o.d" failed with exit status 1

As we can see, the installation out of the box is failing. More specifically, the command

clang -Wno-unused-result -Wsign-compare -Wunreachable-code -fno-common -dynamic -DNDEBUG -g -fwrapv -O3 -Wall -Wstrict-prototypes compile options: '-I/usr/local/include -I['/Users/myself/.virtualenvs/basemap-virtualenv/lib/python3.6/site-packages/numpy/core/include'] ...

The reason lies in a bug on setup.py.  If you read the error, you will see how we are trying to include (-I) a path, but instead, we are passing a python array with a single string inside.  To fix this go to this line:

geos_include_dirs = [os.path.join(GEOS_dir,'include'),inc_dirs]

and change it to:

geos_include_dirs = [os.path.join(GEOS_dir,'include'),inc_dirs[0]]

After that python setup.py install succeeds.